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Localization in weak bialgebras and Hopf envelopes Bennoun, Steve 2013

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Localization in Weak Bialgebras and Hopf Envelopes by  Steve Bennoun A thesis submitted in partial fulfilment of the requirements for the degree of  Doctor of Philosophy in  The Faculty of Graduate Studies (Mathematics)  The University of British Columbia (Vancouver)  April 2013 c Steve Bennoun, 2013  Abstract This dissertation primarily investigates the following questions. Given a weak bialgebra H, is it possible to invert some elements and still have a weak bialgebra structure? In such a case, are there requirements on H or on the set of elements we want to invert? We answer these questions by establishing sufficient conditions for the localization to exist. For instance, the monoid G of all group-like elements in a weak bialgebra H such that G is almost central, i.e. ag = gIg (a), and such that Ig (G) ⊂ G, two conditions we introduce in this thesis, forms a suitable set to be localized. We give a constructive proof of the existence of the localization and detail its weak bialgebra structure. We also prove that it satisfies a universal property. We show these results without any requirement of centrality nor regularity for the elements we invert. We use this work to construct interesting examples of bialgebras and weak bialgebras. We show for instance that GLq (2) has a universal property. We also build the localization of the weak bialgebra associated with a finite directed graph. The examples we exhibit moreover show how our construction can be used in full generality and that our assumptions such as the non-regularity or non-centrality of the elements we invert are not superfluous.  ii  We furthermore present and reformulate Manin’s Hopf envelope. This leads us to define the notion of weak Hopf envelope of a weak bialgebra, and then discuss its relationship with the localization of a weak bialgebra at the monoid of all group-like elements. Finally, in the first part of this thesis we set up an inventory of technical results with detailed proofs for weak bialgebras and coquasi-triangular weak bialgebras. Some of these results are not published in any article and are fully presented here as a basis for future reference.  iii  Preface Part of the material presented in Sections 2.2 and 2.3 was used to produce : Steve Bennoun and Hendryk Pfeiffer, Weak Bialgebras of Fractions, arXiv:math/1212.5775 [math.QA], 2012. The identification of the research program leading to the material of Sections 2.2 and 2.3 was made by Hendryk Pfeiffer and I conducted the research activities. Both authors contributed to the preparation of the manuscript.  iv  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 Fundamental Notions . . . . . . . . . . . . . . 1.1 Coquasi-Triangular Weak Hopf Algebras . . 1.2 Examples . . . . . . . . . . . . . . . . . . . 1.2.1 Mq (2), GLq (2) and SLq (2) . . . . . . 1.2.2 Groupoid Algebra Associated with S3 1.3 Technical Results about WBAs . . . . . . . 1.4 String Diagrams . . . . . . . . . . . . . . . .  1  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  3 3 11 11 14 17 28  2 Localization in a Coquasi-Triangular WBA . 2.1 Ring of Fractions . . . . . . . . . . . . . . . . 2.2 Localization in Coquasi-Triangular Bialgebras 2.2.1 Construction of the Localization . . . . 2.2.2 Example : GLq (2) . . . . . . . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  34 35 38 38 57  v  . . . . . .  . . . . . .  62 64 70 70 82 85  3 Hopf Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Manin’s Hopf Envelope . . . . . . . . . . . . . . . . . . . . . 3.2 Reformulation and Examples . . . . . . . . . . . . . . . . . . 3.2.1 Categorical Reformulation of Manin’s Theorem . . . 3.2.2 Example : Hopf Envelope of Mq (2) . . . . . . . . . . 3.2.3 Example : Hopf Envelope of a Monoid Algebra . . . 3.3 Weak Hopf Envelope and Relationship with the Localization  . . . . . . .  88 88 91 92 97 104 106  2.3  2.2.3 Example : Sweedler’s 4-dimensional Hopf Algebra 2.2.4 Example : Monoid Algebra . . . . . . . . . . . . Localization in a Coquasi-Triangular WBA . . . . . . . . 2.3.1 Construction of the Localization . . . . . . . . . . 2.3.2 Example : Central Group-like Element . . . . . . 2.3.3 Example : Weak Bialgebra of a Directed Graph .  . . . . . .  Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111  vi  List of Figures 1.1  Action groupoid GS3 . . . . . . . . . . . . . . . . . . . . . . . 15  vii  Acknowledgements There are many people to whom I want to express my gratitude for their support and help over the course of these past years. I thank my advisor Patrick Brosnan for giving me the opportunity to undertake this research and for his assistance throughout my degree. I also want to thank Jim Carrell for his help as a committee member and his assistance with administrative matters. Lastly I want to express my sincere gratitude to Hendryk Pfeiffer for introducing me to weak bialgebras, for his constant support and guidance, and for the numerous interesting discussions we have had. I am moreover grateful to my family and friends for their help, and to my wife who has always been there to cheer me up in bad times and make good times better. Finally, I thank the many people I have forgotten to mention but who will undoubtedly recognize themselves here.  viii  Introduction Weak bialgebras and weak Hopf algebras were introduced by B¨ohm, Nill and Szlachanyi in [BNS99] as a generalization of the well-know notions of bialgebras and Hopf algebras. Since then much research has been conducted on this topic. In particular, it has been shown that every fusion category (that is, a k-linear autonomous finitely semi-simple monoidal category with finite-dimensional Hom-spaces that satisfy a specific condition) is equivalent to a category of modules over a weak Hopf algebra ([Hay, Thm. 4.1], [Ost03, Thm. 4] and [ENO05, Cor. 2.22]). Having been developed fairly recently, many questions about weak bialgebras and weak Hopf algebras remain open. In this dissertation, we are interested in the following problem. Given a weak bialgebra, how can we invert, or localize, group-like elements? In particular, what conditions should the weak bialgebra and group-like elements satisfy in order for the localization to make sense and have a weak bialgebra structure? And when the localization exists, what are its properties? In order to answer these questions, we use, as often with weak bialgebras, methods originating in the theory of bialgebras. We give a constructive proof of the existence of the localization, which enables us to study its properties and compute concrete examples. We also investigate Manin’s Hopf envelope. After presenting Manin’s  1  original result, we reformulate it using categorical language, which enables us to apply this construction to any bialgebra. We then define the notion of weak Hopf envelope and show that in specific cases it coincides with the localization relative to the monoid of all group-like elements, thus providing an alternative description of the localization. This thesis is organized as follows. In Chapter 1 we start by introducing weak bialgebras and weak Hopf algebras and prove some of their basic properties. We then establish an inventory of technical results that are needed in subsequent chapters. We next briefly introduce string diagrams. In Chapter 2 we first look at the localization of coquasi-triangular bialgebras and examine some examples. We then establish the localization of a weak bialgebra relative to a suitable subset of group-like elements. Chapter 3 is devoted to the construction of the Hopf envelope. First, we present Manin’s original result. We then reformulate it for any bialgebra using categorical language. Finally, we define the notion of weak Hopf envelope and study its relationship with the localization.  2  Chapter 1 Fundamental Notions In this chapter we introduce the main concepts that will be used throughout this thesis. We also present some basic examples and, in Section 1.3, give some technical results that we shall need in the following chapters. Finally, we present string diagrams and explain how they are used for computations.  1.1  Coquasi-Triangular Weak Hopf Algebras  Definition 1.1.1. A weak bialgebra (H, µ, η, ∆, ε) over a field k is a vector space H such that 1. (H, µ, η) forms an associative algebra with multiplication µ : H ⊗ H → H and unit η : k → H, 2. (H, ∆, ε) forms a coassociative coalgebra with comultiplication ∆ : H → H ⊗ H and counit ε : H → k, 3. the following compatibility conditions hold : • Multiplicativity of the Comultiplication : ∆ ◦ µ = (µ ⊗ µ) ◦ (idH ⊗ σH,H ⊗ idH ) ◦ (∆ ⊗ ∆), 3  (1.1)  • Weak Multiplicativity of the Counit : ε ◦ µ ◦ (µ ⊗ idH ) = (ε ⊗ ε) ◦ (µ ⊗ µ) ◦ (idH ⊗ ∆ ⊗ idH ) = (ε ⊗ ε) ◦ (µ ⊗ µ) ◦ (idH ⊗ ∆op ⊗ idH ), (1.2) • Weak Comultiplicativity of the Unit : (∆ ⊗ idH ) ◦ ∆ ◦ η = (idH ⊗ µ ⊗ idH ) ◦ (∆ ⊗ ∆) ◦ (η ⊗ η) = (idH ⊗ µop ⊗ idH ) ◦ (∆ ⊗ ∆) ◦ (η ⊗ η), (1.3) where σV,W : V ⊗ W → W ⊗ V : v ⊗ w → w ⊗ v flips the two tensor factors. Moreover µop = µ ◦ σH,H is the opposite multiplication and ∆op = σH,H ◦ ∆ is the opposite comultiplication. We also implicitly use Mac Lane’s coherence theorem for the monoidal category Vect [Mac71, Chap. VII], identifying (U ⊗ V ) ⊗ W ∼ = U ⊗ (V ⊗ W ) as well as V ⊗ k ∼ =V ∼ = k⊗V. A homomorphism of weak bialgebras ϕ : H → H is a homomorphism of both unital algebra and counital coalgebra. Definition 1.1.2. Let H be a weak bialgebra. The linear map εs = (idH ⊗ ε) ◦ (idH ⊗ µ) ◦ (σH,H ⊗ idH ) ◦ (idH ⊗ ∆) ◦ (idH ⊗ η)  (1.4)  is called the source counital map whereas εt := (ε ⊗ idH ) ◦ (µ ⊗ idH ) ◦ (idH ⊗ σH,H ) ◦ (∆ ⊗ idH ) ◦ (η ⊗ idH )  (1.5)  is called the target counital map. Their images Hs := εs (H) and Ht := εt (H) form algebras and are respectively called the source base algebra and target base algebra.  4  Remark 1.1.3. The name weak bialgebra is fairly self explanatory. In particular, we see that it is the compatibility between the algebra and coalgebra structures that is weakened. In contrast to a bialgebra, the multiplicativity of the counit ε◦µ=ε⊗ε and the comultiplicativity of the unit ∆◦η =η⊗η do not hold in general anymore and are replaced by (1.2) and (1.3) respectively. Moreover, the condition ε ◦ η = 1k is absent. A weak bialgebra is a bialgebra if and only if εs = η ◦ ε, and if and only if εt = η ◦ ε. Remark 1.1.4. Note that if H is a finite-dimensional weak bialgebra then so is H ∗ . We say that the definition is “self-dual”. Notation 1.1.5. In what follows we use Sweedler’s notation for the coproduct. Namely, let H be a weak bialgebra and a ∈ H, we then write ∆(a) = a ⊗ a as an abbreviation of ai1 ⊗ ai2  ∆(a) = i  with some ai1 , ai2 ∈ H. In other words, with Sweedler’s notation the summation symbol and indices are implied but not explicitly written. Definition 1.1.6. Let (H, µ, η, ∆, ε) be a weak bialgebra over a field k. It is called coquasi-triangular if there exists a linear form r : H ⊗ H → k called the universal r-form that satisfies the following conditions :  5  i) For all a, b ∈ H r(a ⊗ b) = ε(a b )r(a ⊗ b ) = r(a ⊗ b )ε(b a ).  (1.6)  ii) The form r has a weak convolution inverse, i.e. there exists r−1 : H ⊗ H → k such that r(a ⊗ b ) r−1 (a ⊗ b ) = ε(ab).  (1.7)  r−1 (a ⊗ b ) r(a ⊗ b ) = ε(ba),  (1.8)  iii) For all a, b, c ∈ H, we have r(a ⊗ b )b a = a b r(a ⊗ b ),  (1.9)  r(ab ⊗ c) = r(b ⊗ c )r(a ⊗ c ),  (1.10)  r(a ⊗ bc) = r(a ⊗ b)r(a ⊗ c).  (1.11)  Note that condition (1.9) implies that the commutativity inside H is “controlled” by the r-form, this is why one often says that H is almost commutative. A homomorphism of coquasi-triangular weak bialgebras ϕ : (H, r) → (H , r ) is a homomorphism of weak bialgebra satisfying r ◦ (ϕ ⊗ ϕ) = r. Remark 1.1.7. A coquasi-triangular weak bialgebra that is a bialgebra is also coquasi-triangular as a bialgebra. In this case, one can simply omit (1.6) since it is automatically satisfied in a bialgebra. Moreover r(a ⊗ b ) r−1 (a ⊗ b ) = ε(a)ε(b) = r−1 (a ⊗ b ) r(a ⊗ b )  (1.12)  since in a bialgebra ε(ab) = ε(a)ε(b) = ε(b)ε(a) = ε(ba). Remark 1.1.8 (Dual of [NTV03, Prop. 5.2]). The category of finite-dimensional 6  comodules over a weak bialgebra is monoidal. If we then consider the category of finite-dimensional comodules over a coquasi-triangular weak bialgebra, it is moreover braided. Lemma 1.1.9. Let (H, r) be a coquasi-triangular weak bialgebra, then the coopposite weak bialgebra (H op,cop , r−1 ) is coquasi-triangular as well. Remark 1.1.10. If we refer to “(1.10)” in the following, this indicates either the direct use of this equality for (H, r) or the use of the corresponding equality r−1 (ab ⊗ c) = r−1 (b ⊗ c )r−1 (a ⊗ c ) for (H op,cop , r−1 ). The context will every time make clear in which situation we are. Definition 1.1.11. A weak Hopf algebra (H, µ, η, ∆, ε, S) is a weak bialgebra (H, µ, η, ∆, ε) with a linear map S : H → H, called the antipode, that satisfies : µ ◦ (S ⊗ idH ) ◦ ∆ = εs ,  (1.13)  µ ◦ (idH ⊗ S) ◦ ∆ = εt ,  (1.14)  S = µ ◦ (µ ⊗ idH ) ◦ (S ⊗ idH ⊗ S) ◦ (∆ ⊗ idH ) ◦ ∆.  (1.15)  A homomorphism of weak Hopf algebras ϕ : H → H is a homomorphism of weak bialgebras. Remark 1.1.12.  i) In this case, the Hopf algebra axioms  µ ◦ (S ⊗ idH ) ◦ ∆ = η ◦ ε  and  µ ◦ (idH ⊗ S) ◦ ∆ = η ◦ ε  are weakened to (1.13) and (1.14) respectively whereas (1.15) is new. ii) The axioms of the previous definition imply that the antipode is both an algebra- and coalgebra anti-homomorphism. In other words, for a weak Hopf algebra H and a, b ∈ H, we have S(ab) = S(b)S(a) and S(a ) ⊗ S(a ) = S(a) ⊗ S(a) . 7  iii) Note that any homomorphism of weak bialgebras ϕ between two weak Hopf algebras (H, S) and (H , S ) automatically satisfies S ◦ ϕ = ϕ ◦ S. This is proved in the next lemma. Lemma 1.1.13. Let ϕ : H → H be a homomorphism of weak bialgebras and let H, H be weak Hopf algebras. Then S ◦ ϕ = ϕ ◦ S. Proof. Let a ∈ H. Using the weak Hopf algebra axioms, we find 1.15  S (ϕ(a)) = S (ϕ(a) )ϕ(a) S (ϕ(a) ) 1.13  = εs (ϕ(a) )S (ϕ(a) ) = εs (ϕ(a ))S (ϕ(a )) = ϕ(εs (a ))S (ϕ(a ))  1.13  = ϕ(S(a )a )S (ϕ(a )) = ϕ(S(a ))ϕ(a )S (ϕ(a )) = ϕ(S(a ))ϕ(a ) S (ϕ(a ) )  1.14  = ϕ(S(a ))εt (ϕ(a )) = ϕ(S(a ))ϕ(εt (a ))  1.14  = ϕ(S(a )a S(a ))  1.15  = ϕ(S(a)),  where uses that ϕ(1) = 1 and ε (ϕ(a)) = ε(a) and thus ϕ(εs (a)) = εs (a) and ϕ(εt (a)) = εt (a). Example 1.1.14. Let k be a field, G = (G0 , G1 ) a groupoid with finite set of objects G0 and f, f ∈ G1 . Consider the free vector space k[G1 ] on the set of morphisms G1 . One can build a weak Hopf algebra structure on k[G1 ] that is called the groupoid algebra associated with G and then denoted k[G].  8  Its structure is given as follows: µ(f ⊗ f ) =  f ◦f 0  η(1) =  idx ,  if target(f ) = source(f ) , otherwise  x∈G0  ∆(f ) = f ⊗ f, ε(f ) = 1  ∀f ∈ G1 ,  S(f ) = f −1 . Note that, due to its construction, a groupoid algebra is always cocommutative. Moreover, this is an example of the weak Hopf algebra that is not a Hopf algebra. Notation 1.1.15. From now on we shall abbreviate weak bialgebra by WBA and weak Hopf algebra by WHA. Moreover, coquasi-triangular will be written “CQT”; thus a coquasi-triangular weak bialgebra will be called CQT WBA. We now introduce a concept that will play an important role in the rest of this thesis. Definition 1.1.16. Let H be a WBA. An element g ∈ H is called right group-like if ∆(g) = g1 ⊗ g1 and εs (g) = 1, (1.16) it is called left group-like if ∆(g) = 1 g ⊗ 1 g  and  εt (g) = 1.  (1.17)  An element g ∈ H is called group-like if it is both right and left group-like. We denote the set of group-like elements of H by G(H). Notation 1.1.17. In what follows we sometimes have two or more units showing up in our computations. In order to differentiate them and keep 9  track of which one is which one, we use subscripts. Hence we have, for example, εs (a) = 1 ε(a1 ) and then 1εs (a) = 11 12 ε(a12 ). Lemma 1.1.18. The set of group-like elements G(H) of a WBA H is a monoid. Proof. i) 1 ∈ H is group-like. We have 11 · 12 ⊗ 13 · 12 = 1 ⊗ 1 = ∆(1) and similarly 11 · 12 ⊗ 11 · 13 = ∆(1). Furthermore, εs (1) = 11 ε(12 ·11 ) = 1 and εt (1) = ε(11 ·12 )11 = 1. ii) If g, h ∈ G(H) then gh ∈ G(H). We have ∆(gh) = (gh) ⊗ (gh) = g h ⊗ g h = (g11 )(12 h) ⊗ (g11 )(12 h) = g(11 12 h) ⊗ g(11 12 h) = g(1 h) ⊗ g(1 h) = g(h1 ) ⊗ g(h1 ) = (gh)1 ⊗ (gh)1 , by definition of the comultiplication; multiplicativity of the comultiplication; definition of group-like; associativity; associativity and unit axiom; definition of group-like; associativity. Similarly, ∆(gh) = 1 (gh) ⊗ 1 (gh). We furthermore have 1.21  εs (gh) = εs (εs (g)h) = ε(1 · h) = ε(h) = 1, where we have used that g is group-like. In a similar way, εt (gh) = 1.  Lemma 1.1.19. Let H be a WHA. Then every group-like element is invertible with g −1 = S(g) and G(H) forms a group. Proof. Let g ∈ H be group-like. Then S(g)g = S(g)1g = S(g)εs (1)g = S(g)S(1 )1 g ∗  = S(1 g)1 g = S(g )g = εs (g) = 1, 10  where ∗ uses that S is an algebra anti-homomorphism. One similarly finds gS(g) = 1 and hence g −1 = S(g). Let us now look at the group structure of G(H). From the previous lemma we know that 1 is group-like and that if g and h are in G(H) then so is gh. It remains to prove that for g group-like, so is g −1 . We have ∆(g −1 ) = (g −1 ) ⊗ (g −1 ) = g −1 g(g −1 ) ⊗ g −1 g(g −1 ) = g −1 g(1g −1 ) ⊗ g −1 g(1g −1 ) = g −1 g1 (g −1 ) ⊗ g −1 g1 (g −1 ) = g −1 (g1 )(g −1 ) ⊗ g −1 (g1 )(g −1 ) = g −1 g (g −1 ) ⊗ g −1 g (g −1 ) = g −1 (gg −1 ) ⊗ g −1 (gg −1 ) = g −1 1 ⊗ g −1 1 , and similarly ∆(g −1 ) = 1 g −1 ⊗ 1 g −1 . Finally, we have 1.20  εs (g −1 ) = εs (1g −1 ) = εs (εs (g)g −1 ) = εs (gg −1 ) = εs (1) = 1, and similarly εt (g −1 ) = 1, hence g −1 is group-like.  1.2  Examples  In this section we introduce some examples of Hopf algebras and WHAs that will be useful later in this dissertation.  1.2.1  Mq (2), GLq (2) and SLq (2)  We present here the construction of Mq (2), GLq (2) and SLq (2). The reader interested in having more details is referred to [Kas95, Chap. IV.3-6 & VIII.7].  11  Definition 1.2.1. Let k be a field and let q ∈ k be a non-zero element such that q 2 = −1. Let Mq (2) be the quotient of the free algebra k{t11 , t12 , t21 , t22 } by the two-sided ideal Jq generated by t12 t11 = qt11 t12 ,  t22 t12 = qt12 t22 ,  t21 t11 = qt11 t21 ,  t22 t21 = qt21 t22 ,  t12 t21 = t21 t12 ,  t11 t22 − t22 t11 = (q −1 − q)t12 t21 .  (1.18)  Remark 1.2.2. When q = 1 the commutative algebra we get is called M (2). This algebra has a natural grading given by the degree of the polynomials and since the ideal Jq is generated by quadratic elements, it induces a grading on Mq (2) for which the generators t11 , t12 , t21 , t22 are of degree one. We shall now see that Mq (2) has a CQT bialgebra structure. Proposition 1.2.3 ([Kas95, Chap. IV.5 & VIII.7]). The algebra Mq (2) has a coquasi-triangular structure with the comultiplication and counit given by 2  tim ⊗ tmj  ∆(tij ) =  and ε(tij ) = δij .  m=1  This type of coalgebra structure is often called matrix coalgebra because it is dual to a matrix algebra. Moreover, this coalgebra structure can be rewritten in the following compact form, where each entry of the matrix gives an equation : ∆  t11 t12 t21 t22  =  t11 t12 t21 t22  =  t11 ⊗ t11 + t12 ⊗ t21 t11 ⊗ t12 + t12 ⊗ t22 t21 ⊗ t11 + t22 ⊗ t21 t21 ⊗ t12 + t22 ⊗ t22  ⊗  12  t11 t12 t21 t22  and ε  t11 t12 t21 t22  =  1 0 0 1  .  The r-form r : Mq (2)⊗Mq (2) → k is given on the generators of Mq (2)⊗Mq (2) by    r    t11 ⊗ t11 t11 ⊗ t12 t21 ⊗ t11 t21 ⊗ t21  t11 ⊗ t12 t21 ⊗ t12 t11 ⊗ t22 t21 ⊗ t22  t12 ⊗ t11 t22 ⊗ t11 t12 ⊗ t21 t22 ⊗ t21  t12 ⊗ t12 t12 ⊗ t22 t22 ⊗ t12 t22 ⊗ t22          =      q 0 0 q − q −1 0 1 0 0  0 1 0 0  0 0 0 q     .    Using (1.10) and (1.11) one checks that the r-form is 0 on the ideal defined by relations (1.18). We can therefore use equations (1.10) and (1.11) to extend the r-form the all Mq (2). We now introduce a special element in Mq (2). Definition 1.2.4. The element detq = t11 t22 − q −1 t12 t21 = t22 t11 − qt12 t21 is called the quantum determinant. Remark 1.2.5. Note that the quantum determinant is both group-like and central (one verifies this by direct computation). Remark 1.2.6. We know that in a Hopf algebra every group-like element is invertible. As we shall see in Section 2.2.2 adding the inverse of detq to Mq (2) is enough, in this specific case, for the existence of the antipode. Definition 1.2.7. We define GLq (2) = Mq (2)[t]/(detq · t − 1) and SLq (2) = Mq (2)/(detq − 1) = GLq (2)/(t − 1).  13  Proposition 1.2.8 ([Kas95, Thm. IV.6.1]). Both GLq (2) and SLq (2) are Hopf algebras with the bialgebra structure inherited from Mq (2) and the antipode given by  S  1.2.2  t11 t12 t21 t22  =  t11 t12 t21 t22  −1  = det−1 q  t22 −qt12 −1 −q t21 t11  .  Groupoid Algebra of the Action Groupoid Associated with S3  In this section we construct the WHA structure on the groupoid algebra of the groupoid generated by the action of the symmetric group on a set of three elements and the related structure on its dual. Let us start by presenting how one can construct the groupoid algebra of the groupoid associated with a group action. ¯ be a group acting on a set S. One defines a Proposition 1.2.9. Let G groupoid G = (G0 , G1 ) in the following way. The set of objects G0 is simply the set S. For two elements s, s ∈ S, there is a morphism from s to s if ¯ such that g · s = s . there is g ∈ G Proof. To prove that G is actually a groupoid, we first have to check that every object has an identity morphism. This follows form the fact that ¯ Then the associativity in the e · s = s where e is the unit element of G. composition of the morphisms follows directly from the fact that for a group action g · (g · s) = (g g) · s. Finally, to have a groupoid, we still need every morphism to be invertible. Let fg : s → s be the morphism associated with the action g · s = s . Then fg−1 : s → s is the inverse morphism; indeed, fg ◦fg−1 = ids and fg−1 ◦fg = ids since g −1 ·(g ·s) = s and g ·(g −1 ·s ) = s . Definition 1.2.10. The groupoid constructed in the previous proposition is ¯ on S. called the action groupoid associated with the action of G 14  (12), (123)  9 •Q j  (12), (132)  (13), (123)  2)  13 ,( 3)  (1  (2 3) ,( 13 2)  id, (23)  1   3  •Y  (2 =•e  id, (13)  (23), (123)  z  id, (12)  Figure 1.1: Action groupoid GS3 We now construct the groupoid algebra of the action groupoid associated with the action by permutation of the symmetric group S3 on the set S = {1, 2, 3}.  Example 1.2.11. Consider the action of S3 on S = {1, 2, 3} and the action groupoid GS3 associated with it. Between any two objects i, j there are two morphisms “in each direction” and two morphisms between i and itself. This is summarized in Figure 1.1 . For i, j, k ∈ S, we denote the morphism from j to i associated to the transposition (ij) by eij : j → i and the one associated to the 3-cycle (123) or (132) by e¯ij : j → i. Similarly, the morphism eii : i → i is associated with (jk) whereas e¯ii : i → i is associated with the identity morphism. Let k be a field; we know from Example 1.1.14 that the groupoid algebra k[GS3 ] of the action groupoid GS3 has a WHA structure. In our case, this  15  structure is given by : µ(eij ⊗ ekl ) = δjk eil ,  µ(¯ eij ⊗ e¯kl ) = δjk eil ,  µ(¯ eij ⊗ ekl ) = δjk e¯il ,  µ(eij ⊗ e¯kl ) = δjk e¯il ,  η(1) = e¯11 + e¯22 + e¯33 , ∆(eij ) = eij ⊗ eij ,  ∆(¯ eij ) = e¯ij ⊗ e¯ij ,  ε(eij ) = 1,  ε(¯ eij ) = 1 ∀i, j ∈ S,  S(eij ) = eji ,  S(¯ eij ) = e¯ji .  As mentioned in Example 1.1.14, this groupoid algebra is cocommutative. Later in this thesis, we shall look at coquasi-triangular weak bialgebras and weak Hopf algebras. Starting with a finite-dimensional cocommutative WHA, one can take the dual to obtain a commutative, and thus trivially coquasi-triangular, weak Hopf algebra. In the present case, since {e11 , e¯11 , . . . , e¯33 } forms a basis of GS3 , its dual {e∗11 , e¯∗11 , . . . , e¯∗33 } forms a basis of k[GS3 ]∗ . Then the WHA structure on k[GS3 ]∗ is given by : multiplication unit comulitplication counit antipode  m(f ⊗ g)(a) = ∆∗ (f ⊗ g)(a) = f (a )g(a ), u(1) = ε∗ (1) = ε, cm(f )(a ⊗ b) = µ∗ (f )(a ⊗ b) = f (ab), cu(f ) = η ∗ (f ) = f (1), S ∗ (f )(a) = f (S(a)),  with f, g ∈ G∗S3 , a, b ∈ GS3 . Note that k[GS3 ]∗ is commutative since by definition of ∆ we have that m(f ⊗ g)(a) = (f ⊗ g)∆(a) = (f ⊗ g)(a ⊗ a) = f (a)g(a)  16  and the field k is of course commutative. Hence k[GS3 ]∗ is coquasi-triangular with the trivial r-form.  1.3  Technical Results about WBAs  In this section we present some technical results we shall use later in this thesis. Many of the results presented here are scattered around the literature while others are used but not published in any paper. Hence, out of completeness, and as a basis for future reference, we prove here the lemmata and propositions we shall need in the next chapters. Lemma 1.3.1. Let H be a WBA, a, b ∈ H. We have εs (1 ) ⊗ 1 = 1 ⊗ 1  and  1 ⊗ εt (1 ) = 1 ⊗ 1 ,  (1.19)  ε(εs (a)b) = ε(ab)  and  ε(aεt (b)) = ε(ab),  (1.20)  εs (εs (a)b) = εs (ab)  and  εt (aεt (b)) = εt (ab).  (1.21)  1.3  Proof. i) We have εs (1 ) ⊗ 1 = 11 ε(12 11 ) ⊗ 12 = 1 ε(1 ) ⊗ 1 = 1 ⊗ 1 , and similarly 1 ⊗ εt (1 ) = 1 ⊗ 1 . 1.2  ii) We have ε(εs (a)b) = ε(1 ε(a1 )b) = ε(1 b)ε(a1 ) = ε(a1b) = ε(ab). We similarly prove that ε(aεt (b)) = ε(ab). iii) Equalities 1.21 are direct consequences of 1.19 and 1.2.  Lemma 1.3.2. Let H be a WBA and a ∈ H. Then εs (a) = 1 ε(εs (a)1 ),  (1.22)  ∆ ◦ εs (a) = 1 ⊗ εs (a)1 ,  (1.23)  1 ⊗ 1 εs (a) = εs (a) ⊗ εs (a) ,  (1.24)  a ⊗ εs (a ) = a1 ⊗ εs (1 ), 17  (1.25)  Proof.  εs (a) ⊗ εs (a) = εs (εs (a) ) ⊗ εs (a) ,  (1.26)  εt (a) ⊗ εt (a) = εt (a) ⊗ εt (εt (a) ).  (1.27)  i) Using (1.2), we have 1 ε(εs (a)1 ) = 12 ε(11 ε(a11 )12 ) = 12 ε(a11 )ε(11 12 ) = ε(a11 12 )12 = 1 ε(a1 ) = εs (a).  ii) Using (1.3) we find ∆ ◦ εs (a) = ∆(1 ε(a1 )) = 1 ⊗ 1 ε(a1 ) = 11 ⊗ 12 11 ε(a12 ) = 1 ⊗ εs (a)1 . iii) Using (1.3) we find 1 ⊗ 1 εs (a) = 11 ⊗ 11 12 ε(a12 ) = 1 ⊗ 1 ε(a1 ) = (1 ) ⊗ (1 ) ε(a1 ) = εs (a) ⊗ εs (a) .  18  iv) We have a ⊗ εs (a ) = a ⊗ 1 ε(a 1 ) = (a11 ) ε((a11 ) 12 ) ⊗ 12 = a 11 ε(a 11 12 ) ⊗ 12 1.2  = a 11 ε(a 11 )ε(11 12 ) ⊗ 12 = a (11 ) ε(a (11 ) )ε(11 12 ) ⊗ 12  = (a11 ) ε((a11 ) )ε(11 12 ) ⊗ 12 = a11 ⊗ 12 ε(11 12 ) = a1 ⊗ εs (1 ). v) Using (1.3) again, we find εs (a) ⊗ εs (a) = 1 ⊗ 1 ε(a1 ) = 11 ⊗ 11 12 ε(a12 ) = 11 ε(11 ) ⊗ 11 12 ε(a12 ) = 11 ε(12 11 ) ⊗ 12 13 ε(a13 ) = εs (11 ) ⊗ 11 12 ε(a12 ) = εs (1 ) ⊗ 1 ε(a1 ) = εs (εs (a) ) ⊗ εs (a) . We prove in a similar way that εt (a) ⊗ εt (a) = εt (a) ⊗ εt (εt (a) ).  Lemma 1.3.3. Let (H, r) be a CQT WBA. Then r(a ⊗ 1) = r(1 ⊗ a) = ε(a).  (1.28)  Proof. We have 1.7  ε(a) = ε(1 · a) = r(1 ⊗ a )r−1 (1 ⊗ a ) = r(11 · 12 ⊗ a )r−1 (11 ⊗ a ) 1.10  1.7  = r(12 ⊗ a )r(11 ⊗ a )r−1 (11 ⊗ a ) = r(12 ⊗ a )ε(11 · a ) = r(1 ⊗ a). 19  Similarly, ε(a) = r(a ⊗ 1). The next lemma will give us the tools required to prove equality (1.42). Lemma 1.3.4. Let H be a WBA and a, b ∈ H. Then  Proof.  εt (a ) ⊗ εs (a ) = εt (a ) ⊗ εs (a ),  (1.29)  εs (a ) ⊗ εt (a ) = εs (a ) ⊗ εt (a ),  (1.30)  εt (a )ε(a b) = εt (ab),  (1.31)  ε(ab )εs (b ) = εs (ab),  (1.32)  εt ((ab) ) ⊗ εs ((ab) ) = εt (a )ε(a b ) ⊗ εs (b ),  (1.33)  εt ((ab) ) ⊗ εs ((ab) ) = εt (a ) ⊗ εs (b )ε(a b ),  (1.34)  εt ((ab) ) ⊗ εs ((ab) ) = ε(a b )εt (a ) ⊗ εs (b ).  (1.35)  i) We have εt (a ) ⊗ εs (a ) = ε(11 a )11 ⊗ 12 ε(a 12 ) 1.2  = ε(11 a )11 ⊗ 12 ε(a 12 ) = εt (a ) ⊗ εs (a ).  Using this, we straightforwardly get that εs (a )⊗εt (a ) = εs (a )⊗εt (a ). ii) Here we have 1.2  εt (a )ε(a b) = ε(1 a )1 ε(a b) = ε(1 ab)1 = εt (ab). We similarly prove that ε(ab )εs (b ) = εs (ab).  20  iii) We have εt ((ab) ) ⊗ εs ((ab) ) = εt (a b ) ⊗ εs (a b ) ♦  = εt (a )ε(a b ) ⊗ ε(a b )εs (b ) = εt (a )ε(a b ) ⊗ εs (b ), where ♦ follows from (1.31) and (1.32). iv) We have 1.33  εt ((ab) ) ⊗ εs ((ab) ) = εt (a )ε(a b ) ⊗ εs (b ) 1.20  = εt (a )ε(a εt (b )) ⊗ εs (b )  1.29  = εt (a )ε(a εt (b )) ⊗ εs (b )  1.20  = εt (a ) ⊗ εs (b )ε(a b ).  We similarly prove that εt ((ab) ) ⊗ εs ((ab) ) = ε(a b )εt (a ) ⊗ εs (b ).  The next lemma will help us prove (1.45). Lemma 1.3.5. Let H be a WBA and a, b ∈ H. Then aεt (b) = ε(a b)a ,  (1.36)  εs (a)b = b ε(ab ),  (1.37)  ε(ac )ε(bc ) = ε(aεt (c) )ε(bεt (c) ),  (1.38)  εt (a ) ⊗ εt (a ) = εt (εt (a) ) ⊗ εt (εt (a) ),  (1.39)  ε(ab )εs (εt (b )) = ε(a b)εs (a ).  (1.40)  If H is moreover CQT with r-form r, then r(a ⊗ b)εs (εt (a )) = r(a ⊗ b )εs (b ).  21  (1.41)  Proof.  i) We have aεt (b) = ε((aεt (b)) )(aεt (b)) = ε((aε(1 b)1 ) )(aε(1 b)1 ) = ε((a1 ) )ε(1 b)(a1 ) = ε(a 1 )ε(1 b)a 1 1.2  = ε(a 1 b)a 1 = ε((a1) b)(a1) = ε(a b)a .  We similarly prove that εs (a)b = b ε(ab ). ii) We have 1.37  1.20  1.37  ε(ac )ε(bc ) = ε(aεs (b)c) = ε(aεs (b)εs (c)) = ε(aεt (c) )ε(bεt (c) ). iii) We have εt (a ) ⊗ εt (a ) = ε(11 a )11 ⊗ ε(12 a )12 1.38  = ε(11 εt (a) )11 ⊗ ε(12 εt (a) )12 = εt (εt (a) ) ⊗ εt (εt (a) ).  iv) We have 1.20  ε(ab )εs (εt (b )) = ε(aεt (b ))εs (εt (b )) 1.39  = ε(aεt (εt (b) ))εs (εt (εt (b) ))  1.20  = ε(aεt (b) )εs (εt (εt (b) ))  1.27  = ε(aεt (b) )εs (εt (b) )  1.32  = εs (aεt (b))  22  1.36  = ε(a b)εs (a ).  v) We have 1.6  r(a ⊗ b)εs (εt (a )) = r(a ⊗ b )ε(b a )εs (εt (a )) 1.40  = r(a ⊗ b )ε(b a )εs (b )  1.6  = r(a ⊗ b )εs (b ).  Lemma 1.3.6. Let (H, r) be a CQT WBA and a, b ∈ H. Then εt (a ) ⊗ εs (b )r(a ⊗ b ) = r(a ⊗ b )εt (b ) ⊗ εs (a ),  Proof.  (1.42)  r(a ⊗ εt (b)) = ε(ab),  (1.43)  r(a ⊗ εs (b)) = ε(ba),  (1.44)  r(εs (a) ⊗ b) = ε(bεs (a)),  (1.45)  r(εt (a) ⊗ b) = ε(εt (a)b),  (1.46)  r−1 (a ⊗ εs (b)) = ε(aεs (b)),  (1.47)  r−1 (εt (a) ⊗ b) = ε(ba).  (1.48)  i) We have 1.6  εt (a ) ⊗ εs (b )r(a ⊗ b ) = εt (a ) ⊗ εs (b )ε(a b )r(a ⊗ b ) 1.34  = εt ((a b ) ) ⊗ εs ((a b ) )r(a ⊗ b )  1.9  = r(a ⊗ b )εt ((b a ) ) ⊗ εs ((b a ) )  1.35  = r(a ⊗ b )ε(b a )εt (b ) ⊗ εs (a )  1.6  = r(a ⊗ b )εt (b ) ⊗ εs (a ).  23  ii) We have r(a ⊗ εt (b)) = r(a ⊗ ε(1 b)1 ) = ε(1 b)r(a ⊗ 1 ) 1.20  = ε(εs (1 )b)r(a ⊗ 1 ) = ε(a )ε(εs (1 )b)r(a ⊗ 1 ) = ε(εt (a ))ε(εs (1 )b)r(a ⊗ 1 )  1.42  = r(a ⊗ 1 )ε(εs (a )b)ε(εt (1 )) = r(a ⊗ 1 )ε(εs (a )b)ε(1 ) = r(a ⊗ 1)ε(εs (a )b)  1.20  1.28  = r(a ⊗ 1)ε(a b) = ε(a )ε(a b)  = ε(ab). Similarly, one has that r(a ⊗ εs (b)) = ε(ba). iii) We have r(εs (a) ⊗ b) = r(1 ε(a1 ) ⊗ b) ♦  = ε(εs (a)εt (1 ))r(1 ⊗ b) = ε(12 ε(a12 )ε(13 11 )13 )r(11 ⊗ b) = ε(13 11 )ε(12 13 )ε(a12 )r(11 ⊗ b) 1.3  = ε(13 11 )ε(13 12 )ε(a12 )r(11 ⊗ b) = ε(ε(13 11 )13 12 ε(a12 ))r(11 ⊗ b) = ε(εt (1 )εs (a))r(1 ⊗ b)  1.20  = ε(εs (εt (1 ))εs (a))r(1 ⊗ b)  1.41  = ε(εs (b )εs (a))r(1 ⊗ b ) = ε(b εs (a))ε(b ) = ε(bεs (a)),  where ♦ follows from (1.19) and (1.20) and from (1.20) and (1.28). We similarly prove that r(εt (a) ⊗ b) = ε(εt (a)b).  24  iv) The equality r−1 (a ⊗ εs (b)) = ε(aεs (b)) follows from Lemma 1.1.9 and (1.46). On the other hand, r−1 (εt (a) ⊗ b) = ε(ba) follows from Lemma 1.1.9 and (1.44) .  Notation 1.3.7. In what follows we often take many times the comultiplication of an element. In order to make it easy to read we slightly modify Sweedler’s notation (cf. Notation 1.1.5); namely, we use roman numbers for “orders” higher than three, thus a becomes aIV whereas a is written aV . Using this convention we have, for example, (∆ ⊗ ∆ ⊗ ∆) ◦ (∆ ⊗ id) ◦ ∆(a) = (∆ ⊗ ∆ ⊗ ∆)(a ⊗ a ⊗ a ) = a ⊗ a ⊗ a ⊗ aIV ⊗ aV ⊗ aV I . We now observe some facts about group-like elements in WBAs that will be useful later in this thesis. Proposition 1.3.8. Let (H, r) be a CQT WBA, a, b ∈ H and g ∈ G(H) a group-like element. Then i) ε(ag) = ε(a),  (1.49)  ii) ∆(ag) = a g ⊗ a g,  (1.50)  iii) r−1 (a ⊗ g)r(a ⊗ g) = ε(a),  (1.51)  iv) r(a ⊗ g)r−1 (a ⊗ g) = ε(a),  (1.52)  v) r(a ⊗ g )r(b ⊗ g ) = r(a ⊗ g)r(b ⊗ g)ε(b a ),  (1.53)  vi) r−1 (a ⊗ g )r−1 (b ⊗ g ) = ε(a b )r−1 (a ⊗ g)r−1 (b ⊗ g).  (1.54)  If (H, r) is a CQT bialgebra, then we moreover have r(a ⊗ g −1 ) = r−1 (a ⊗ g) and r(g −1 ⊗ a) = r−1 (g ⊗ a).  (1.55)  Proof. In this proof, we indicate by ∗ the equalities where we use that g is 25  group-like. i) We have 1.20  ε(ag) = ε(aεt (g)) = ε(a · 1) = ε(a). ii) We find ∆(ag) = a g ⊗ a g = a 1 g ⊗ a 1 g = (a1) g ⊗ (a1) g = (a ⊗ a )(g ⊗ g) = a g ⊗ a g. iii) We have r−1 (a ⊗ g)r(a ⊗ g) = r−1 (a ⊗ g)ε((a 1) )ε((a 1) )r(aIV ⊗ g) = r−1 (a ⊗ g)ε(a 1 )ε(a 1 )r(aIV ⊗ g) 1.43  = r−1 (a ⊗ g)ε(a 1 )r(a ⊗ εt (1 ))r(aIV ⊗ g)  1.19  = r−1 (a ⊗ g)ε(a 1 )r(a ⊗ 1 )r(aIV ⊗ g)  1.11  = r−1 (a ⊗ g)ε(a 1 )r(a ⊗ 1 g)  1.19  = r−1 (a ⊗ g)ε(a εs (1 ))r(a ⊗ 1 g)  1.47  = r−1 (a ⊗ g)r−1 (a ⊗ εs (1 ))r(a ⊗ 1 g)  1.19  = r−1 (a ⊗ g)r−1 (a ⊗ 1 )r(a ⊗ 1 g)  1.11  = r−1 (a ⊗ 1 g)r(a ⊗ 1 g) ∗  = r−1 (a ⊗ g )r(a ⊗ g ) 1.8  1.20  ∗  = ε(ga) = ε(εs (g)a) = ε(1a)  = ε(a).  iv) Similar to iii).  26  v) Here we have ∗  r(a ⊗ g )r(b ⊗ g ) = r(a ⊗ g1 )r(b ⊗ g1 ) 1.11  = r(a ⊗ g)r(a ⊗ 1 )r(b ⊗ g)r(b ⊗ 1 )  1.10  = r(a ⊗ g)r(b ⊗ g)r(b a ⊗ 1)  1.28  = r(a ⊗ g)r(b ⊗ g)ε(b a ).  vi) Similar to v). vii) First, note that 1.11  r(a ⊗ g)r(a ⊗ g −1 ) = r(a ⊗ gg −1 ) = r(a ⊗ 1) = ε(1 · a) = ε(a), and that using 1.11 we similarly find that r(a ⊗ g −1 )r(a ⊗ g) = ε(a). Then ∗  r−1 (a ⊗ g) = r−1 (a ⊗ g)ε(a g) = r−1 (a ⊗ g)ε(a )ε(g) ∗  = r−1 (a ⊗ g)ε(a ) = r−1 (a ⊗ g)r(a ⊗ g)r(a ⊗ g −1 ) 1.8  = ε(ga )r(a ⊗ g −1 ) = ε(g)ε(a )r(a ⊗ g −1 ) ∗  = r(a ⊗ g −1 ), where ∗ uses that g is group-like. The other equality is proved in a similar fashion.  27  1.4  String Diagrams  In this section we briefly introduce a very convenient tool for depicting morphisms in any monoidal category called “string diagrams”. Here we want to apply this tool to the structure maps µ, η, ∆, ε, and S of WBAs and WHAs. We therefore work in the monoidal category Vectk where k is a field. One of the great advantages of this method is that it fully uses the extension for symmetric monoidal categories of Mac Lane’s coherence theorem [Mac71, Chap. VII]. In other words, many computations are made easier because this notation implicitly uses the fact that we can identify (U ⊗ V ) ⊗ W ∼ = U ⊗ (V ⊗ W ) as well as k ⊗ V ∼ =V ∼ = V ⊗ k. As we shall see later in this section, using string diagrams makes it particularly easy to see when two elements are equal and which properties or equalities can be used. String diagrams, even though widely used, do not often appear in published articles. One “historical” reason is that it used to be technically difficult for publishers to smoothly insert them in written texts. More importantly these diagrams take lots of space and thus considerably expand the length of papers; this is the main reason why we mostly use equations and not diagrams in this thesis. Due to their widespread use and importance, we nonetheless introduce them in an informal manner, focusing on how they are used, and show a few computations. The reader interested in a formal and systematic introduction to string diagrams is referred to [JS91] and [JS]. Remark 1.4.1. When working in a monoidal category C there are two main ways of producing new morphisms from old ones; by composing them or by taking their tensor product. Suppose that in C we have the following objects and morphisms a : A → B ⊗ B,  b : B → C ⊗ D,  c : B ⊗ C → C, 28  d : D ⊗ C → D.  It is sometimes unclear whether two expressions such as f1 = (idB ⊗ c ⊗ d) ◦ (idB ⊗ idB ⊗ b ⊗ idC ) ◦ (a ⊗ idB ⊗ idC )(A ⊗ B ⊗ C) and f2 = (idB ⊗ idC ⊗ d) ◦ (idB ⊗ c ⊗ idD ⊗ idC ) ◦ (a ⊗ b ⊗ idC )(A ⊗ B ⊗ C) are equal or not. When drawing their diagrams we obtain  a  a b c  and d  b c d  for f1 and f2 respectively. The convention is to use lines for objects and circles with labels for morphisms (the lines are only labeled if there is a possible ambiguity). We moreover read them from top to bottom and a blank space between two objects or morphisms represent a tensor product. We see that these two diagrams are a “deformation” of each other and thus we intuitively want them to represent the same element. Now that we have had a first glance at string diagrams and understand the basic idea behind them, let us see more precisely how we construct them. Construction 1.4.2. We are going to define string diagrams as classes of graphs satisfying specific properties. Let a < b be real numbers. A plane graph (between levels a and b) is a topological graph Γ embedded in R × [a, b] such that every elements x ∈ Γ ∩ {a, b} are nodes and that they belong to the closure of exactly one edge. The 29  nodes on level a are called the domain whereas the nodes on level b are the codomain. By convention these nodes (called outer nodes) are not represented on diagrams. Since we want to read these diagrams from top to bottom, we consider oriented plane graphs. We moreover don’t want turn backs and thus require the graphs to be progressive, i.e. the projection on the vertical axis p2 : R × [a, b] → [a, b] is injective on each edge. As mentioned above we want to be able to deform the diagrams without changing the elements they represent. A deformation of progressive plane graph is a continuous function h : Γ × [0, 1] → R × [a, b] such that for each t ∈ [0, 1] the function h(−, t) : Γ → R × [a, b] is a homeomorphism. We then say that two progressive plane graphs are equivalent if there exists a deformation between them. We thus consider classes of progressive plane graphs under this equivalence relation. If the domain of a diagram Γ equals the codomain of Ω we can compose them and the diagram of Γ◦Ω is simply the juxtaposition of the two diagrams with Ω on top, appropriately connected to Γ at the bottom. The tensor product Γ ⊗ Ω of two diagrams Γ and Ω is formed by putting them side by side (on the same level). Many conventions are employed when using string diagrams. Here we summarize the most important ones when working with a weak bialgebra (H, µ, η, ∆, ε) over a field k. Convention 1.4.3. • Domain and Codomain. As already indicated the nodes of the domain and codomain (which are simply the identity morphism on one element) are not indicated. • The Field k. The field is not indicated in the diagrams, in other words we implicitly identify k ⊗ H ∼ =H∼ = H ⊗ k.  30  • Unit and counit. The unit is depicted by  whereas the counit is depicted by  . • Multiplication and Comultiplication. In order to make string diagrams more readable, the multiplication is pictured by  µ  instead of  and the comultiplication by  instead of  ∆ .  As an example, we are going to prove equation (1.25) using string diagrams. Before that, let us see how some of the basic WBA axioms of Definition 1.1.1 are expressed in term of diagrams. Remark 1.4.4. Let (H, µ, η, ∆, ε) be a WBA over a field k and let a, b, c ∈ H. Let us see how we depict some axioms using string diagrams (these will be used in the following proof): 31  SD1 The unit axiom 1a = a = a1 is represented by  =  = .  SD2 The counit axiom ε(a ) ⊗ a = a = a ⊗ ε(a ) is represented by  =  = .  SD3 The coassociativity axiom (a ) ⊗ (a ) ⊗ a = a ⊗ (a ) ⊗ (a ) is represented by  = . SD4 The multiplicativity of the comultiplication (ab) ⊗ (ab) = a b ⊗ a b is represented by  = . Note that there is no vertex in the middle of the right-hand side picture. SD5 The weak multiplicativity of the counit ε(abc) = ε(ab )ε(b c) = ε(ab )ε(b c) is depicted by 32  =  = .  We are now ready to prove equation (1.25). Lemma 1.4.5. Let H be a WBA and x ∈ H. Then x ⊗εs (x ) = x1 ⊗εs (1 ). Proof. Indeed, we have  SD1  =  =  εs  SD4  SD5  SD3  =  =  =  SD4  SD2  =  =  = εs ,  where  follows from the definition of εs .  33  Chapter 2 Localization in a Coquasi-Triangular WBA In this chapter we shall introduce the localization of a WBA relative to a monoid of group-like elements. More precisely, starting with a CQT WBA (H, r) and G a sub monoid of the set of group-like elements G(H), we want to construct the WBA H[G−1 ] and WBA homomorphism ϕ : H → H[G−1 ] having the following universal property. For any WBA homomorphism ψ : ¯ such that ψ(g) is invertible for any g ∈ G, there exists a unique H → H ¯ such that the following diagram WBA homomorphism ψ¯ : H[G−1 ] → H commutes ϕ / H[G−1 ] H #   ψ  ψ¯  ¯ H.  This generalizes the well-known notion of localization of a non-commutative ring R relative to a multiplicatively closed subset S.  34  2.1  Ring of Fractions  The main sources we use for this section are [Ste75, Chap. II.1] and [Row88, Chap. 3.1]. Many other sources exist but the advantage of these ones is to prove the results in the most general case; in particular the set S can contain non-regular elements.  Definition 2.1.1. Let R be a ring and S ⊂ R a subset. We call S multiplicatively closed if s, t ∈ S implies st ∈ S  and  1 ∈ S.  Definition 2.1.2. Let R be a ring and S be a multiplicatively closed set. We define the right ring of fractions of R with respect to S as a ring R[S −1 ] together with a ring homomorphism ϕ : R → R[S −1 ] satisfying : • ϕ(s) is invertible for any s ∈ S, • every element in R[S −1 ] is of the form ϕ(a)ϕ(s)−1 with a ∈ R, s ∈ S, • a ∈ ker ϕ if and only if as = 0 for some s ∈ S. Left rings of fractions are defined in a similar way. Rings of fractions have the following universal property. Proposition 2.1.3 ([Ste75, Prop. II.1.1.1]). Suppose R[S −1 ] exists, then for ¯ such that ψ(s) is invertible for every every ring homomorphism ψ : R → R ¯ such that the s ∈ S, there exists a unique homomorphism ψ¯ : R[S −1 ] → R following diagram commutes R  ϕ  /  R[S −1 ] "   ψ  ψ¯  ¯ R.  35  Corollary 2.1.4. When it exists, a ring of fractions R[S −1 ] is unique up to a unique isomorphism. As one would expect, the ring of fractions does not always exist. We shall now see a condition characterizing its existence. Proposition 2.1.5 ([Ste75, Prop. II.1.1.4] or [Row88, Thm. 3.1.4]). Let R be a ring and S multiplicatively closed set of R. i) The ring of fractions R[S −1 ] exists if and only if S satisfies : (S1) if s1 ∈ S and a1 ∈ R, there exist s2 ∈ S and a2 ∈ R such that s 1 a2 = a1 s 2 , (S2) if s1 a = 0 for s1 ∈ S, a ∈ R, then there exists s2 ∈ S such that as2 = 0. ii) When R[S −1 ] exists, it is of the form R[S −1 ] = R × S/ ∼ where ∼ is the equivalence relation for which (a1 , s1 ) ∼ (a2 , s2 ) if there exist u1 , u2 ∈ R such that a1 u1 = a2 u2 ∈ R  and  s1 u1 = s2 u2 ∈ S. Then, writing is given by  a s  (2.1) (2.2)  for the equivalence class containing (a, s), the addition a1 a2 a1 v1 + a2 v2 + = s1 s2 s1 v1  (2.3)  where condition S1 ensures the existence of v1 ∈ S and v2 ∈ R such that s1 v1 = s2 v2 ∈ S. Furthermore, the multiplication is given by a1 a2 a1 v1 · = s1 s2 s2 v2 36  (2.4)  where S1 is used again to find v1 ∈ R and v2 ∈ S such that s1 v1 = a2 v2 . Remark 2.1.6. i) From the previous proposition we see that condition S1 ensures that two fractions can always be put on the same denominator. They then represent the same class if one can find a common denominator for which the numerators are equal. ii) The condition S2 will be nearly exclusively used in the following way. Let s1 ∈ S and a1 , a2 ∈ R be such that s1 a1 = s1 a2 . Then we have 0 = s1 a1 − s1 a2 = s1 (a1 − a2 ) and thus by condition S2 there exists s2 ∈ S such that (a1 − a2 )s2 = 0. Hence we have an element s2 ∈ S, such that a1 s2 = a2 s2 . We can therefore reformulate condition S2 as follows : (S2’) if s1 a1 = s1 a2 with s1 ∈ S and a1 , a2 ∈ R, there exists s2 ∈ S such that a1 s2 = a2 s2 . Indeed, suppose that S2’ holds and that s1 a = 0 with s1 ∈ S and a ∈ R. Then, we have in particular that s1 a = 0 = s1 0 and thus by S2’ there exists s2 ∈ S such that a1 s2 = 0s2 = 0. Thus we have just proved that conditions S2 and S2’ are equivalent. iii) Note that the definitions of the addition (2.3) and multiplication (2.4) are independent of the choice of v1 and v2 . Definition 2.1.7. A multiplicatively closed set S satisfying the conditions S1 and S2 of the preceding Proposition is called a right denominator set or right Ore set in reference to [Ore31]. Example 2.1.8 (Commutative Case). Let R be a commutative ring and S a multiplicatively closed set. In this case both conditions S1 and S2 are trivially satisfied by taking s2 = s1 and a2 = a1 . An important special case is when one takes S to be the set of all regular elements (i.e. non-zero-divisors) in R; then R[S −1 ] is called the classical (right) ring of fractions or total (right) ring of fractions. 37  2.2  Localization in Coquasi-Triangular Bialgebras  In the first part of this section we follow [Hay92] and use the theory presented above to construct the localization of a coquasi-triangular bialgebra at a sub monoid of group-like elements. In order to make the generalization of this construction as easy as possible, we reorganize the order and complete many proofs of the above reference.  2.2.1  Construction of the Localization  Let us introduce a definition that will play a central role in our construction. Definition 2.2.1. Let (B, r) be a coquasi-triangular bialgebra and let g be a group-like element of B. We define the linear map Ig : B → B by Ig (a) = r(a ⊗ g)a r−1 (a ⊗ g) for any a ∈ B. Lemma 2.2.2. The linear map Ig defined in the previous definition is a CQT bialgebra homomorphism. Proof. First, using relation (1.10), the fact that g is a group-like and Lemma 1.1.9, we have Ig (ab) = r((ab) ⊗ g)(ab) r−1 ((ab) ⊗ g) 1.10  = r(a ⊗ g)r(b ⊗ g)a b r−1 (a ⊗ g)r−1 (b ⊗ g) = Ig (a)Ig (b).  38  Moreover, the unit 1 being group-like we get Ig (1) = r(1 ⊗ g)1r−1 (1 ⊗ g) = 1ε(1 · g) = 1, where we have used that g is group-like. Let us now check that Ig is a coalgebra homomorphism. We have (Ig ⊗ Ig )(∆(a)) = r(a ⊗ g)a r−1 (a ⊗ g) ⊗ r(aIV ⊗ g)aV r−1 (aV I ⊗ g) = r(a ⊗ g)a ε(ga ) ⊗ aIV r−1 (aV ⊗ g) = r(a ⊗ g)a ⊗ a r−1 (aIV ⊗ g) = ∆(Ig (a)), and ε(Ig (a)) = r(a ⊗ g)ε(a )r−1 (a ⊗ g) = r(a ⊗ g)r−1 (a ⊗ g) = ε(ag) = ε(a)ε(g) = ε(a). Thus Ig is a bialgebra homomorphism. Using properties of the r-form we find 1.10  r(Ig (a) ⊗ Ig (b)) = r(a b ⊗ g)r(a ⊗ b )r−1 (b a ⊗ g) = r(a b r(a ⊗ b ) ⊗ g)r−1 (b a ⊗ g) 1.9  = r(a ⊗ b )r(b a ⊗ g)r−1 (b a ⊗ g) = r(a ⊗ b )ε(b a g) = r(a ⊗ b )ε(b )ε(a )ε(g) = r(a ⊗ b),  and hence Ig is a CQT bialgebra homomorphism. 39  Proposition 2.2.3. i) For each g ∈ G(B), the endomorphism Ig is a CQT bialgebra automorphism; its inverse being given by −1 I−1 g (a) = r (a ⊗ g)a r(a ⊗ g).  ii) Let G ⊂ G(B) be a monoid of group-like elements. The morphism I : G → Aut(B) : g → Ig is a monoid anti-homomorphism. is the inverse of Ig ; it Proof. i) It is straightforward to verify that I−1 g then follows directly that I−1 g is also a CQT bialgebra homomorphism and thus Ig is an automorphism. ii) This follows from (1.11) and Lemma 1.3.3.  We now give some useful properties of Ig . Proposition 2.2.4. Let (B, r) be a CQT bialgebra, a, b ∈ B, g, h ∈ G(B). Then i) gIg (a) = ag,  (2.5)  I−1 g (a)g = ga.  (2.6)  Ig (h) = h,  (2.7)  ii)  iii) ga = 0 if and only if ag = 0, iv) the monoid G(B) is commutative. 40  Proof.  i) We have gIg (a) = r(a ⊗ g)ga r−1 (a ⊗ g) 1.9  = a g r(a ⊗ g)r−1 (a ⊗ g) = a gε(a g)  = a gε(a )ε(g) = ag. Using the definition of I−1 and 1.9 again, one similarly finds that g I−1 g (a)g = ga. ii) Using that h is group-like, we get Ig (h) = r(h ⊗ g)hr−1 (h ⊗ g) = hε(hg) = h. iii) Suppose ga = 0 then 2.7  2.5  0 = Ig (ga) = Ig (g)Ig (a) = gIg (a) = ag. Similarly ag = 0 implies ga = 0. iv) This follows directly from i) and ii).  This Proposition enables us to prove the next Lemma 2.2.5. Let (B, r) be a CQT bialgebra and G be a sub monoid of G(B). Then seeing B as a ring and G as a multiplicatively closed set, they satisfy conditions S1 and S2 of Prop. 2.1.5. Proof. Using the notations of Prop. 2.1.5, let s1 = g ∈ G and a1 = a ∈ B. By (2.5) we have gIg (a) = ag, i.e. a2 = Ig (a) and s2 = g. Concerning condition S2, suppose ga = 0. Then ag = 0 by Prop. 2.2.4 iii), i.e. we can simply take 41  s2 = g.  Remark 2.2.6. This result is important since it will enable us to define an equivalence relation ∼ on B × G as well as a ring structure on (B × G)/ ∼ in analogy to Prop. 2.1.5. We shall, out of completeness, give all the details of the proof that (B, G)/ ∼ forms an algebra. We now have all the tools required to prove the following Theorem 2.2.7. Let (B, r) be a CQT bialgebra and G a sub monoid of G(B). Define the equivalence relation ∼ on B × G as follows : (a1 , g1 ) ∼ (a2 , g2 ) if there exist u1 , u2 ∈ B such that a1 u1 = a2 u2 ∈ B  and  (2.8)  g1 u1 = g2 u2 ∈ G.  (2.9)  Let us denote by ag the equivalence class containing (a, g). Then B[G−1 ] := (B × G)/ ∼ has a unital algebra structure given by a + g a · g where  1 1  b ah + bg = , h gh b aIg (b) = , h gh  is the unit and scalar multiplication is given by α · ag =  (2.10) (2.11) αa g  for α ∈ k.  In order to make the proof easy to read, we are going to break it down into small parts. Lemma 2.2.8. Let (B, r), G and ∼ be as in Thm. 2.2.7. Then the relation ∼ is indeed an equivalence relation. Proof. For the reflexivity just take u1 = u2 = 1 and for the symmetry simply exchange u1 and u2 . 42  Let us now prove the transitivity of ∼. Suppose there exist u1 , u2 , v2 , v3 such that a1 u1 = a2 u2 g1 u1 = g2 u2  and  a1 g1  ∼  a2 g2  and  a2 g2  ∼  a3 , g3  i.e.  a2 v2 = a3 v3 g2 v2 = g3 v3 .  Consider g2 u2 ∈ G and g2 v2 ∈ G; by S1(which is satisfied by Lem. 2.2.5) there exist w1 , w2 such that g2 u2 w1 = g2 v2 w2 ∈ G. Then by S2’ there exists h ∈ G such that u2 w1 h = v2 w2 h. (2.12) Hence 2.12  a1 u1 w1 h = a2 u2 w1 h = a2 v2 w2 h = a3 v3 w2 h  and  2.12  g1 u1 w1 h = g2 u2 w1 h = g2 v2 w2 h = g3 v3 w2 h, with g1 u1 w1 h belonging to G since both g2 u2 w1 and h do. We now prove a technical result that will enable us to substantially simplify subsequent proofs. Lemma 2.2.9. Let (B, r) and G be as in Thm. 2.2.7. Then ag11 ∼ ag22 if and only if there exist v1 , v2 ∈ G (and not simply in B) such that a1 v1 = a2 v2 and g1 v1 = g2 v2 ∈ G. Proof. If we have v1 and v2 in G such that a1 v1 = a2 v2 and g1 v1 = g2 v2 ∈ G then clearly ag11 ∼ ag22 . Now suppose that ag11 ∼ ag22 ; we have to show that we can find v1 , v2 in G such that a1 v1 = a2 v2 and g1 v1 = g2 v2 ∈ G. By definition there exist u1 , u2 ∈ B such that a1 u1 = a2 u2  and g1 u1 = g2 u2 ∈ G. 43  By equations (2.5) and (2.7) we get u1 g1 = g1 Ig1 (u1 ) = Ig1 (g1 u1 ) = g1 u1 ; similarly g2 u2 = u2 g2 . Hence v1 := u1 g1 g2 and v2 := u2 g1 g2 belong to G. Then a1 v1 = a1 u1 g1 g2 = a2 u2 g1 g2 = a2 v2 and g1 v1 = g1 u1 g1 g2 = g2 u2 g1 g2 = g2 v2 ∈ G. Remark 2.2.10. Note that the definition of the equivalence relation only requires v1 , v2 to be in B; the previous lemma tells us that these elements can actually be taken in G. This property is known to hold if the elements we invert are central or are all regular [Bea99, Lem. A.5.3]. Thus the previous lemma shows that it is enough for these elements to be “almost central”, i.e. to commute up to a homomorphism, for this property to hold. Proof of Thm. 2.2.7. i) The addition is well-defined. a2 b1 a1 Let g1 ∼ g2 and h1 ∼ hb22 ; by Lemma 2.2.9 there exist u1 , u1 , v1 , v2 ∈ G such that a1 u 1 = a2 u 2 b1 v1 = b2 v2 and g1 u1 = g2 u2 h1 v1 = h2 v2 . We want to show that b1 a1 h1 + b1 g1 a2 h2 + b2 g2 a2 b2 a1 + = ∼ = + . g1 h1 g1 h1 g2 h2 g2 h2 We have (a1 h1 + b1 g1 )u1 v1 = a1 h1 u1 v1 + b1 g1 u1 v1 = a1 u1 h1 v1 + b1 v1 g1 u1 = a2 u2 h2 v2 + b2 v2 g2 u2 = a2 h2 u2 v2 + b2 g2 u2 v2 = (a2 h2 + b2 g2 )u2 v2 , where we have used the commutativity of G (cf. Prop. 2.2.4.iv). More-  44  over we have that (g1 h1 )u1 v1 = g1 u1 h1 v1 = g2 u2 h2 v2 = (g2 h2 )u2 v2 ∈ G. ii) One verifies in a straightforward manner that B[G−1 ] is a vector space. iii) The multiplication is well-defined. As above, take ag11 ∼ ag22 and hb11 ∼ hb22 ; we want to show that a1 b 1 a1 Ig1 (b1 ) a2 Ig2 (b2 ) a2 b2 · = ∼ = · . g1 h1 g1 h1 g2 h2 g2 h2 Then 2.5  a1 Ig1 (b1 )u1 v1 = a1 u1 Iu1 (Ig1 (b1 ))v1 = a2 u2 Ig1 u1 (b1 )v1 2.7  = a2 u2 Ig1 u1 (b1 )Ig1 u1 (v1 ) = a2 u2 Ig1 u1 (b1 v1 ) = a2 u2 Ig2 u2 (b2 v2 ) = a2 u2 Ig2 u2 (b2 )v2 = a2 u2 Iu2 (Ig2 (b2 ))v2 = a2 Ig2 (b2 )u2 v2 ,  where  follows from Prop. 2.2.3.ii). Moreover, we have (g1 h1 )u1 v1 = g1 u1 h1 v1 = g2 u2 h2 v2 = (g2 h2 )u2 v2 ∈ G,  where we have obviously used the commutativity of the monoid G(B). iv) One verifies in a straightforward way the compatibility of the multiplication with the scalars.  v) The multiplication is distributive.  45  We have a g  b c + h k  = = = = =  a bk + ch aIg (bk + ch) · = g hk ghk aIg (bk) + aIg (ch) 2.7 aIg (b)k + aIg (c)h = ghk ghk aIg (b)gk + aIg (c)gh (aIg (b)k + aIg (c)h)g = ghkg ghgk aIg (b) aIg (c) + gh gk a b a c · + · . g h g k  Left distributivity is proved in a similar way. vi) The multiplication is associative. Indeed, we have a b · g h  aIg (b) c aIg (b)Igh (c) c = · = k gh k ghk aI (b)I (c) aI ∗ g hg g (bIh (c)) = = ghk ghk a bIh (c) = · g hk a b c · = , g h k  where ∗ uses the commutativity of G(B). vii) The element Indeed,  1 1  is the unit. a 1 aIg (1) 2.7 a · 1 a · = = = , g 1 g·1 g g  where we have used that 1 is group-like. Similarly,  46  1 1  ·  a g  = ag .  Before building the CQT bialgebra structure of B[G−1 ], let us introduce a notation. Notation 2.2.11. Let B be a CQT bialgebra, G a monoid of group-likes and B[G−1 ] its localization. We denote by ρ the homomorphism given by a ρ : B → B[G−1 ] : a → . 1 Theorem 2.2.12. Let (B, r) be a CQT bialgebra and G a sub monoid of G(B). Then B[G−1 ] has a coquasi-triangular bialgebra structure. Its algebra structure is given by Thm. 2.2.7 whereas the comultiplication, counit and r-form are given by a g a ε g a b ⊗ g h ∆  r  =  a a ⊗ , g g  (2.13)  = ε(a),  (2.14)  = r(a ⊗ b )r−1 (a ⊗ h)  (2.15)  · r−1 (g ⊗ b )r(g ⊗ h). Remark 2.2.13. Note that we could prove this theorem by using the fact that the localization has an algebra structure (Thm. 2.2.7) and then exploit its universal property to build a CQT bialgebra structure. Such a proof would use that the morphisms ∆ and ε are algebra homomorphism, a property that does not hold in the weak case. Such a proof would thus not generalize to the weak case and we therefore do not use this argument here. Proof of Thm. 2.2.12. i) The comultiplication is well-defined. a1 a2 Let g1 ∼ g2 , i.e. there exist u1 , u2 ∈ G such that a1 u 1 = a2 u 2 g1 u1 = g2 u2 . 47  Applying ∆ on both sides of a1 u1 = a2 u2 we get a1 u1 ⊗ a1 u1 = a2 u2 ⊗ a2 u2 . Next applying ρ ⊗ ρ on both sides of the latter, we find a1 u1 a1 u1 a u2 a u2 ⊗ = 2 ⊗ 2 . 1 1 1 1  (2.16)  Now we have ∆  a1 g1  a1 a1 a u1 a u1 ⊗ = 1 ⊗ 1 g1 g1 g1 u1 g1 u 1 1 a1 u1 a1 u1 ⊗ = ⊗ 1 1 g1 u1 a2 u2 a2 u2 1 2.16 ⊗ = ⊗ 1 1 g2 u2 a a a u2 a u2 = 2 ⊗ 2 = 2⊗ 2 g2 u2 g2 u2 g2 g2 a2 . = ∆ g2 =  1 g1 u1 1 g2 u2  ii) The counit is well-defined. Again take ag11 ∼ ag22 ; we have ε  a1 g1  = ε(a1 ) = ε(a1 ) · 1 = ε(a1 )ε(u1 ) = ε(a1 u1 ) = ε(a2 u2 ) = ε(a2 )ε(u2 ) = ε(a2 ) · 1 = ε(a2 ) a2 =ε . g2  iii) The comultiplication is coassociative. ∗ Let a ∈ B, we have (id ⊗ ∆) ◦ (∆(a)) = a ⊗ (a ) ⊗ (a ) = (a ) ⊗ (a ) ⊗a = (∆⊗id)◦(∆(a)) by coassociativity of B. Then by applying ρ ⊗ ρ ⊗ ρ on both sides of ∗, we find a (a ) (a ) (a ) (a ) a ⊗ ⊗ = ⊗ ⊗ . 1 1 1 1 1 1 48  (2.17)  Now we have a (a ) (a ) ⊗ ⊗ g g g a (a ) (a ) = ⊗ ⊗ 1 1 1 (a ) (a ) a 2.17 = ⊗ ⊗ 1 1 1 (a ) (a ) a = ⊗ ⊗ g g g a = (∆ ⊗ id) ◦ ∆ g  a g  (id ⊗ ∆) ◦ ∆  =  1 1 1 ⊗ ⊗ g g g 1 1 1 ⊗ ⊗ g g g  .  iv) The comultiplication is additive. We have ∆  a b + g h  where we have used in  ah + bg gh ah a h bg b g ⊗ + ⊗ = gh gh gh gh a a b b = ⊗ + ⊗ g g h h a b =∆ +∆ , g h =∆  that G is commutative.  v) The counit is additive. Indeed, ε  a b + g h  ah + bg = ε(ah + bg) gh = ε(a)ε(h) + ε(b)ε(g) = ε(a) + ε(b) b a =ε +ε , g h =ε  49  where we have used that ε(g) = 1 = ε(h). vi) One verifies using straightforward computations that the comultiplication and counit are k-linear maps. vii) B[G−1 ] is a counital coalgebra. It remains to show that (id ⊗ ε) ◦ ∆ = id = (ε ⊗ id) ◦ ∆. We have (id ⊗ ε) ◦ ∆  a g  = (id ⊗ ε) =  a a a ε(a )a ⊗ = ε(a ) = g g g g  a , g  where we have used the counit axiom in B. The other equality is proved in a similar way.  We now have to verify the axioms regarding to the compatibility of the algebra and coalgebra structures. viii) The comultiplication preserves the multiplication. We have to prove that ∆ ◦ µ = (µ ⊗ µ) ◦ (id ⊗ σ ⊗ id) ◦ (∆ ⊗ ∆), where σ : B[G−1 ] ⊗ B[G−1 ] → B[G−1 ] ⊗ B[G−1 ] : ag ⊗ hb → hb ⊗ ag is the morphism flipping two adjacent tensor factors. We get ∆◦µ  where 2.2.2.  a b ⊗ g h  aIg (b) (aIg (b)) (aIg (b)) = ⊗ gh gh gh a Ig (b) a Ig (b) a Ig (b ) a Ig (b ) = ⊗ = ⊗ gh gh gh gh a b a b = · ⊗ · g h g h a b = (µ ⊗ µ) ◦ (id ⊗ σ ⊗ id) ◦ (∆ ⊗ ∆) ⊗ , g h =∆  holds since Ig is CQT bialgebra homomorphism by Lemma  50  ix) The counit preserves the multiplication. Indeed, ε  where again  a b · g h  aIg (b) = ε(aIg (b)) gh = ε(a)ε(Ig (b)) = ε(a)ε(b) a b ε , =ε g h  =ε  holds since Ig is CQT bialgebra homomorphism.  x) The comultiplication preserves the unit. Indeed, 1 1 1 1 1 = ⊗ ∆ = ⊗ 1 1 1 1 1 since 1 is group-like in B. xi) The counit preserves the unit. Indeed, 1 ε 1  = ε(1) = 1k .  Hence we have just proved that B[G−1 ] is a bialgebra. We now want to show it has an r-form. xii) The r-form is well-defined. Let ag11 ∼ ag22 and hb11 ∼ hb22 , i.e. there exist u1 , u2 , v1 , v2 ∈ G such that a1 u 1 = a2 u 2 g1 u1 = g2 u2  and  b1 v1 = b2 v2 h1 v1 = h2 v2 .  Then using that u1 , u2 , v1 , v2 are group-likes and that the monoid of group-like elements G(B) is commutative by Proposition 2.2.4 iv), we 51  have b1 a1 ⊗ = g1 h1 = r(a1 ⊗ b1 )r−1 (a1 ⊗ h1 )r−1 (g1 ⊗ b1 )r(g1 ⊗ h1 )  r  −1 = r(a1 ⊗ b1 )r−1 (aIV 1 ⊗ h1 )r (g1 ⊗ b1 )r(g1 ⊗ h1 )  · r(a1 ⊗ v1 )r−1 (a1 ⊗ v1 ) r−1 (g1 ⊗ v1 )r(g1 ⊗ v1 ) 1  = r(a1 ⊗ b1 )r(a1 ⊗ v1 )r  −1  (aIV 1  1 −1  ⊗ h1 )r (a1 ⊗ v1 )  · r−1 (g1 ⊗ v1 )r−1 (g1 ⊗ b1 )r(g1 ⊗ v1 )r(g1 ⊗ h1 ) 1.11  = r(a1 ⊗ b1 v1 )r−1 (a1 ⊗ h1 v1 )r−1 (g1 ⊗ b1 v1 )r(g1 ⊗ h1 v1 )  = r(a1 ⊗ (b1 v1 )IV ) r(u1 ⊗ (b1 v1 ) )r−1 (u1 ⊗ (b1 v1 ) ) r−1 (g1 ⊗ (b1 v1 ) ) 1  · r−1 (a1 ⊗ h1 v1 ) r−1 (u1 ⊗ h1 v1 )r(u1 ⊗ h1 v1 ) r(g1 ⊗ h1 v1 ) 1 1.10  −1  = r(a1 u1 ⊗ (b1 v1 ) )r (g1 u1 ⊗ (b1 v1 ) )r−1 (a1 u1 ⊗ h1 v1 )r(g1 u1 ⊗ h1 v1 )  = r((a1 u1 ) ⊗ (b1 v1 ) )r−1 (g1 u1 ⊗ (b1 v1 ) )r−1 ((a1 u1 ) ⊗ h1 v1 )r(g1 u1 ⊗ h1 v1 ) = r((a2 u2 ) ⊗ (b2 v2 ) )r−1 (g2 u2 ⊗ (b2 v2 ) )r−1 ((a2 u2 ) ⊗ h2 v2 )r(g2 u2 ⊗ h2 v2 ) = r(a2 u2 ⊗ (b2 v2 ) )r−1 (g2 u2 ⊗ (b2 v2 ) )r−1 (a2 u2 ⊗ h2 v2 )r(g2 u2 ⊗ h2 v2 ) 1.10  = r(a2 ⊗ (b2 v2 )IV ) r(u2 ⊗ (b2 v2 ) )r−1 (u2 ⊗ (b2 v2 ) ) r−1 (g2 ⊗ (b2 v2 ) ) 1 −1  −1  · r (a2 ⊗ h2 v2 ) r (u2 ⊗ h2 v2 )r(u2 ⊗ h2 v2 ) r(g2 ⊗ h2 v2 ) 1 −1  −1  = r(a2 ⊗ b2 v2 )r (g2 ⊗ b2 v2 )r (a2 ⊗ h2 v2 )r(g2 ⊗ h2 v2 ) 1.11  = r(a2 ⊗ b2 ) r(a2 ⊗ v2 ) r−1 (g2 ⊗ b2 ) r−1 (g2 ⊗ v2 ) ♦ −1  · r (a2 ⊗ v2 ) r  −1  (aIV 2  ⊗ h2 )r(g2 ⊗ h2 ) r(g2 ⊗ v2 ) ♦  −1  −1  = r(a2 ⊗ b2 )r (g2 ⊗ b2 )r (a1 ⊗ h2 )r(g2 ⊗ h2 ) = r(a2 ⊗ b2 )r−1 (a1 ⊗ h2 )r−1 (g2 ⊗ b2 )r(g2 ⊗ h2 )  52  =r  b2 a2 ⊗ g2 h2  ,  and ♦ indicate the terms that “cancel” each other.  where  xiii) Equation 1.11 holds for B[G−1 ]. Indeed, we have a b c a bIh (c) ⊗ · =r ⊗ g h k g hk −1 = r(a ⊗ (bIh (c)) )r (a ⊗ hk)r−1 (g ⊗ (bIh (c)) )r(g ⊗ hk) r  ∗  = r(a ⊗ b Ih (c ))r−1 (a ⊗ kh)r−1 (g ⊗ b Ih (c ))r(g ⊗ hk) 1.11  = r(a ⊗ b ) r(a ⊗ Ih (c )) r−1 (a ⊗ h) r−1 (aIV ⊗ k) ♦ −1  −1  ♦  · r (g ⊗ Ih (c )) r (g ⊗ b )r(g ⊗ h)r(g ⊗ k),  (2.18)  ♦  where ∗ uses that Ih is a bialgebra homomorphism and that G(B) is commutative. Now considering the three terms indicated by ♦, we get r−1 (g ⊗ Ih (c ))r(a ⊗ Ih (c ))r−1 (a ⊗ h) = = r−1 (g ⊗ Ih (c )) r−1 (g ⊗ h)r(g ⊗ h) 1 −1  · r (a ⊗ h)r(a ⊗ h) r(aIV ⊗ Ih (c ))r−1 (aV ⊗ h) 1 1.11  −1  = r (g ⊗ hIh (c ))r(g ⊗ h)r−1 (a ⊗ h) · r(a ⊗ hIh (c ))r−1 (aIV ⊗ h)  2.5  = r−1 (g ⊗ c h)r(g ⊗ h)r−1 (a ⊗ h) · r(a ⊗ c h)r−1 (aIV ⊗ h)  1.11  = r−1 (g ⊗ c ) r−1 (g ⊗ h)r(g ⊗ h) r−1 (a ⊗ h) 1 IV  · r(a ⊗ c ) r(a  ⊗ h)r−1 (aV ⊗ h) 1  53  = r−1 (g ⊗ c )r−1 (a ⊗ h)r(a ⊗ c ). Hence, plugging these terms back into 2.18, we get 2.18 = r(a ⊗ b ) r−1 (a ⊗ h) r(a ⊗ c ) r−1 (aIV ⊗ k) ♦  ♦  −1  −1  · r (g ⊗ c ) r (g ⊗ b )r(g ⊗ h)r(g ⊗ k) ♦  = r(a ⊗ b )r−1 (a ⊗ h)r−1 (g ⊗ b )r(g ⊗ h) · r(a ⊗ c )r−1 (aIV ⊗ k)r−1 (g ⊗ c )r(g ⊗ k) a b a c =r ⊗ r ⊗ g h g k  xiv) We similarly have that r  a b c · ⊗ g h k  =r  b c ⊗ h k  r  a c ⊗ g k  .  xv) Equation 1.9 holds for B[G−1 ], i.e. B[G−1 ] is almost commutative. First recall that we have r−1 (x ⊗ y )x y = y x r−1 (x ⊗ y ) by Lemma 1.1.9 and thus for x = g ∈ B and y = b ∈ G, we get r−1 (g ⊗ b )gb = b gr−1 (g ⊗ b ). Then applying ρ : B → B[G−1 ] : a → a1 on both sides of this equality, it becomes b g −1 gb r−1 (g ⊗ b ) = r (g ⊗ b ) 1 1 which can be rewritten r−1 (g ⊗ b )  g b b g · = · r−1 (g ⊗ b ). 1 1 1 1  54  Now, multiplying on both the right and the left by r  1 b ⊗ g 1  1 g  and noting that  = r(1 ⊗ (b ) )r−1 (1 ⊗ 1)r−1 (g ⊗ (b ) )r(g ⊗ 1) = ε((b ) )ε(1)r−1 (g ⊗ (b ) )ε(g) = r−1 (g ⊗ b ),  we find  1 b ⊗ g 1  r  b 1 1 b · = · r 1 g g 1  1 b ⊗ g 1  .  (2.19)  Similarly one gets for a ∈ B, h ∈ G r  a 1 ⊗ 1 h  1 a a 1 · = · r h 1 1 h  a 1 ⊗ 1 h  ,  (2.20)  moreover using equation 1.9 one has r  b a ⊗ 1 1  a a b b · = · r 1 1 1 1  a b ⊗ 1 1  .  (2.21)  Then, we get a b b a ⊗ · = g h h g a 1 b 1 b a · ⊗ · · =r 1 g 1 h h g 1 b 1 a b 1 b a 1.10 = r ⊗ · r ⊗ · · g 1 h 1 1 h h g 1 b 1 1 a b a 1 1.11 = r ⊗ r ⊗ r ⊗ r ⊗ g 1 g h 1 1 1 h  r  b 1 a 1 · · · 1 h 1 g 2.20  2.20 2.20  = r  1 b ⊗ g 1  r  1 1 ⊗ g h  r  a b ⊗ 1 1 2.21  55  r  a 1 ⊗ 1 h  b a 1 1 · · · 1 1 h g 2.21  2.21  = r  1 b ⊗ g 1  r  1 1 ⊗ g h  a b ⊗ 1 1  r  r  a 1 ⊗ 1 h  a b 1 1 · · · 1 1 g h  a 1 ⊗ 1 h  a 1 b 1 · · · 1 g 1 h  2.19  2.19  1 1 a b 1 b ⊗ r ⊗ r ⊗ r = r g 1 g h 1 1 1 b 1 a b 1 a b =r ⊗ · r ⊗ · · g 1 h 1 1 h g h a 1 b 1 a b =r · ⊗ · · 1 g 1 h g h b a b a ⊗ · , =r g h g h  2.19  where we have used that Ig (1) = 1 = Ih (1).  1 g  1 h  and  commute since g and h do and that  Proposition 2.2.14. Let (B, r) and G ⊂ G(B) be as above. Then ρ : B → B[G−1 ] : a → a1 is a CQT bialgebra homomorphism and B enjoys the following universal property: for any CQT bialgebra homomorphism ψ : ¯ such that ψ(g) is invertible for any g ∈ G, there exists a unique CQT B→B ¯ such that bialgebra homomorphism ψ¯ : B[G−1 ] → B B  ρ  /  B[G−1 ] #   ψ  ψ¯  ¯ B.  commutes. Proof. Using Lemma 1.3.3 it is straightforward to verify that ρ is a CQT bialgebra homomorphism. Then ψ¯ is defined by ψ¯ ag = ψ(a)ψ(g)−1 and is obviously uniquely determined by ψ. We then verify by direct computation that ψ¯ is bialgebra homomorphism. The fact that ψ¯ preserves the r-from directly follows from (1.55).  56  Remark 2.2.15. Note that the homomorphism ρ : B → B[G−1 ] satisfies the conditions of Definition 2.1.2. In other words, B[G−1 ] is the ring of fractions of B with respect to G (on which we have built a CQT bialgebra structure). Corollary 2.2.16. When the localization exists, it is unique up to a unique isomorphism.  2.2.2  Example : GLq (2)  Let us now apply this construction to the coquasi-triangular bialgebra Mq (2) introduced in section 1.2.1. Example 2.2.17. Let G be the monoid generated by the group-like element detq = t11 t22 − q −1 t12 t21 = t22 t11 − qt12 t21 . The coquasi-triangular bialgebra Mq (2)[G−1 ] exists and its structure is given, for a, b ∈ Mq (2) and g, h ∈ G, by b ah + bg = , h gh b ab · = , h gh a a a ∆ = ⊗ , g g g a g a g  r  a b ⊗ g h  +  0 , 1 1 the unit is : , 1 a ε = ε(a) , g the zero is :  = r(a ⊗ b )r−1 (a ⊗ h)r−1 (g ⊗ b )r(g ⊗ h).  (2.22) (2.23) (2.24)  (2.25)  Remark 2.2.18. i) Note that the multiplication in GLq (2) as defined above follows from Remark 2.1.6 iii) and the fact that the monoid G is in the centre of Mq (2) since detq is central. Indeed, using the centrality of g we have gb = bg and thus using (2.4) in Proposition 2.1.5 we find that a b ab · = . g h gh  57  ii) Recall from Definition 1.2.7 that GLq (2) = Mq (2)[t]/(detq t − 1). It is this latter description that we are going to use in the proof of the next proposition. Proposition 2.2.19. The coquasi-triangular bialgebras GLq (2) and Mq (2)[G−1 ] are isomorphic. Remark 2.2.20. Note that one could prove that GLq (2) is isomorphic to the localization Mq (2)[G−1 ] by verifying that GLq (2) satisfies the universal property of the localization. Nevertheless, as our goal is here to illustrate the construction of the localization presented in the previous section, we are going to use explicitly construct an isomorphism between GLq (2) and Mq (2)[G−1 ]. Proof of Prop. 2.2.19. In order to prove this, we are going to define two homomorphisms inverse of each other. First, for a ∈ Mq (2), let us define ϕ : GLq (2) −→ Mq (2)[G−1 ] a a −→ 1 1 t −→ . detq Then note that since the monoid G is generated by detq , any g ∈ G is of the form α · detnq with α ∈ k and n ∈ N0 . We can thus define ψ : Mq (2)[G−1 ] −→ GLq (2) a −→ a 1 1 −→ tn . n detq We now prove that ϕ and ψ are well-defined bialgebra homomorphisms and inverse of each other.  58  i) ϕ is well-defined. Indeed, 1 detq 1 − 1 detq 1 detq 1 1 1 0 = − = − = . detq 1 1 1 1  ϕ(detq t − 1) = ϕ(detq )ϕ(t) − ϕ(1) =  ii) ϕ is linear. Let α, β ∈ k and a, b ∈ Mq (2). Then αa + βb αa1 + βb1 αa βb a b = = + =α +β 1 1·1 1 1 1 1 = αϕ(a) + βϕ(b).  ϕ(αa + βb) =  iii) ϕ is an algebra homomorphism. First, note that ϕ(1) = 11 . Moreover ϕ(a)ϕ(b) =  ab a b · = = ϕ(ab) 1 1 1  and similarly ϕ(a)ϕ(tn ) = ϕ(atn ). iv) ϕ is a coalgebra homomorphism. First, we have ε(ϕ(a)) = ε a1 = ε(a). Furthermore ∆(ϕ(a)) = ∆  a a a = ⊗ = (ϕ ⊗ ϕ) ◦ ∆(a). 1 1 1  v) ϕ preserves the r-form.  59  Indeed, b a m ⊗ detq detnq n m = r(a ⊗ b )r−1 (a ⊗ detnq )r−1 (detm q ⊗ b )r(detq ⊗ detq )  r(ϕ(atm ) ⊗ ϕ(btn )) = r  1.55  = r(a ⊗ b )r(a ⊗ tn )r(tm ⊗ b )r(tm ⊗ tn )  = r(atm ⊗ btn ), where the last equality follows from (1.10) and (1.11). vi) ψ is well-defined. Let detam ∼ detb n , by Lemma 2.2.9 there exist u1 , u2 ∈ G such that q  q  au1 = bu2  and  n detm q u1 = detq u2 .  Since u1 , u2 ∈ G we know that there exist r, s ∈ N0 such that u1 = detrq and u2 = detsq . Using the above equalities we get adetrq = bdetsq  and  r n s detm q detq = detq detq .  Since {ti11 tj12 tk21 tl22 }i,j,k,l 0 forms a basis of Mq (2) by [Kas95, Thm. IV.4.1], it follows that m + r = n + s. We then have ψ  b detnq  = btn = bdetsq tn+s = adetrq tn+s = atn+s−r = atm = ψ  vii) ψ is linear. This immediately follows from the construction of ψ. viii) ψ is an algebra homomorphism.  60  a detm q  .  Indeed, we have ψ  a b m · detq detnq  =ψ  a detm q  ψ  Moreover, we have ψ  1 1  =ψ  ab detm+n q  b detnq  .  = abtm+n = atm btn  = 1.  ix) ψ is a coalgebra homomorphism. Using that t is group-like, we have (ψ ⊗ ψ) ◦ ∆  a detm q  = (ψ ⊗ ψ)  a a m ⊗ detq detm q  = a tm ⊗ a tm = ∆(a)∆(tm ) = ∆(atm ) = ∆ ◦ ψ  a detm q  .  Using again that t is group-like, we find for the counit that ε◦ψ  a detm q  = ε(atm ) = ε(a)ε(tm ) = ε(a) = ε  a detm q  .  x) ψ preserves the r-form. Similar to v). xi) ψ ◦ ϕ = idGLq (2) . Indeed, we have ψ(ϕ(a)) = ψ  xii) ϕ ◦ ψ = idMq (2)[G−1 ] . We have ϕ◦ψ  a = a and ψ(ϕ(tm )) = ψ 1  a detm q  = ϕ(atm ) =  61  1 detm q  a 1 a . m = 1 detq detm q  = tm .  Hence, we have thus just proved that ϕ and ψ are bialgebra isomorphisms.  Remark 2.2.21. As noted in Remark 2.2.18 i) the centrality of the monoid G implies that we don’t have to use the morphism Idetq when defining the multiplication in Mq (2)[G−1 ]. Note that this fact alone doesn’t mean that Ig is the identity. In the present case we do have Idetq = idMq (2) and this follows from both centrality and regularity of the elements of G. Indeed, then for any g ∈ G and a ∈ Mq (2) we have by (2.5) that gIg (a) = ag, and thus gIg (a) − ag = 0. Then, by centrality of g we get g(Ig (a) − a) = 0. Since g is non-zero and not a zero-divisor (since Mq (2) itself has no zero-divisor) this forces Ig (a) = a. In the next example we shall take a monoid G that is regular but not central and see that Ig is in general not the identity.  2.2.3  Example : Sweedler’s 4-dimensional Hopf Algebra  This Hopf algebra was introduced by Sweedler in his seminal book [Swe69]. Theorem 2.2.22. Let H4 be the algebra generated by two elements g and y subject to the relations g 2 = 1,  y 2 = 0 and  gy = −yg.  Then H4 has a Hopf algebra structure if we define the comultiplication ∆ by ∆(g) = g ⊗ g  ∆(y) = y ⊗ 1 + g ⊗ y,  and  the counit ε by ε(g) = 1  and 62  ε(y) = 0,  and the antipode S by S(g) = g −1  and  S(y) = yg.  Moreover, note that {1, g, y, yg} is a basis of the underlying vector space. As noted for example by Majid, it turns out that this Hopf algebra has a coquasi-triangular structure. Proposition 2.2.23 ([Maj95, p. 51]). Sweedler’s 4-dimensional Hopf algebra H4 is coquasi-triangular; its r-form is given by    r    1 ⊗ 1 1 ⊗ g 1 ⊗ y 1 ⊗ yg g ⊗ 1 g ⊗ g g ⊗ y g ⊗ yg y ⊗ 1 y ⊗ g y ⊗ y y ⊗ yg yg ⊗ 1 yg ⊗ g yg ⊗ y yg ⊗ yg          =      1 1 0 1 −1 0 0 0 α 0 0 −α  0 0 α α     ,    where α ∈ k is arbitrary. Remark 2.2.24. The only group-likes in H4 being 1 and g, the monoid of group-like elements is G = {1, g}. Note that since gy = −yg, the monoid G is not in the centre of H4 . The elements of G are nonetheless regular. Indeed, let α1 , α2 , α3 , α4 , β1 , β2 , β3 , β4 ∈ k and suppose that g(α1 1 + α2 g + α3 y + α4 yg) = g(β1 1 + β2 g + β3 y + β4 yg), then α2 1 + α1 g − α4 y − α3 yg = β2 1 + β1 g − β4 y − β3 yg and thus α 1 = β1 ,  α2 = β2 ,  α3 = β3 ,  α4 = β4 ,  since {1, g, y, yg} is a basis of the underlying vector space. Hence we have just proved that ga = gb implies a = b for any a, b ∈ H4 and thus g is left63  regular. We similarly show that g is right-regular and therefore regular. To sum up, our monoid is in this case regular but not central. Remark 2.2.25. In order to construct the localization H4 [G−1 ] we need to compute I1 and Ig . The morphism I1 is the identity by Lemma 1.3.3. Then using equation (1.12) to compute r−1 and the definition of Ig (Def. 2.2.1) we get Ig (1) = 1, Ig (g) = g, Ig (y) = −y, Ig (yg) = −yg. We thus see that in this case Ig = idH4 . Finally note that since the elements of G are already invertible in H4 , we get that H4 ∼ = H4 [G−1 ]. The main purpose of this example is not so much to compute the localization H4 [G−1 ] as to show that Ig is not always the identity.  2.2.4  Example : Monoid Algebra  In this example we are going to construct the localization of the monoid algebra k[M ] at the monoid generated by all basis vectors. One special feature of this example is that the bialgebra k[M ] has non-regular group-like elements; this will be established in Remark 2.2.29. Let us first note the following property of monoids. Lemma 2.2.26. Let (M, ·, 1) be a finite monoid. Let g ∈ M and ϕ : M → M : a → ga the multiplication on the left by g. Then g is invertible if and only if ϕ is a bijection of sets. Proof. First assume that g is invertible. Then the map ψ : M → M : a → g −1 a is the inverse of ϕ and thus ϕ is bijective. Now assume that ϕ is bijective. Then ϕ−1 (1) exists and is unique. It turns 64  out that this element is the inverse of g. Indeed, gϕ−1 (1) = ϕ(ϕ−1 (1)) = 1. Moreover ϕ(ϕ−1 (1)g) = gϕ−1 (1)g = 1g = g = ϕ(1) which implies, by injectivity of ϕ, that ϕ−1 (1)g = 1. Corollary 2.2.27. Let M be a finite monoid that is not a group. Then there exist g, h, k ∈ M such that h = k and gh = gk. Proof. Indeed, M being not a group, there exists a non-invertible element g ∈ M . Then, by the preceding lemma, the left multiplication by g is not bijective. The monoid M being finite, this amounts to saying that it is not injective, i.e. there exist h, k ∈ M such that h = k and gh = gk.  Proposition 2.2.28. Let (M, ·, 1) be a finite monoid that is not a group, then the monoid algebra k[M ] has a bialgebra structure. Denoting by e1 , ea , eb the basis vectors of k[M ] associated with 1, a, b ∈ M , the bialgebra structure is given by µ(ea ⊗ eb ) = eab ,  ∆(ea ) = ea ⊗ ea ,  η(1) = e1 ,  ε(ea ) = 1.  Proof. This checked by direct computations. Remark 2.2.29. Let M be a monoid that is not a group. Then, in contrast to the previous examples, the bialgebra k[M ] has non-regular group-likes. Indeed, let g be a non-invertible element in M . By Corollary 2.2.27 there exist h and k different from each other and such that gh = gk. Therefore eg (eh − ek ) = egh − egk = 0 even though both eg and eh − ek are non-zero. The element eg is thus a group-like and a zero-divisor. We now want to construct the localization of k[M ]. In order to do so, we need a coquasi-triangular structure. 65  Let us take the monoid M to be commutative, then so is k[M ], i.e. k[M ] is CQT with a trivial r-form. Then taking G = {ea | a ∈ M } to be the monoid of group-like elements generated by the ea ’s, we can then construct the localization k[M ][G−1 ] using Theorem 2.2.7. Its bialgebra structure is given, for a, b, c, d ∈ M , by  ea eb ea eb  ec ea ed + ec eb = , ed eb ed ec ea ec · = , ed eb ed ea ea ea ⊗ , = ∆ eb eb eb +  0 , e1 e1 the unit is : , e1 ea ε = ε(ea ). eb the zero is :  The natural question that arises here is whether this localization is isomorphic to another object we know. The answer to this question is in the next Proposition 2.2.30. Let M be as above and let M be the group defined by generators and relations as follows. The set of generators is {a | a ∈ M } ∪ {a | a ∈ M } and the relations {aa = 1 = aa | a ∈ M }. Then the bialgebras k[M ] ∼ = k[M ][G−1 ] are isomorphic. Proof. Let ϕ : k[M ][G−1 ] → k[M ] be defined by ϕ to show that this is an isomorphism.  ea eb  = ea e¯b . We are going  i) ϕ is well-defined. Suppose eeab ∼ eedc , i.e. there exist eu , ev ∈ G such that ea eu = ec ev  and  66  eb eu = ed ev .  Then ϕ  ea eb  = ea e¯b = ea eu eu¯ e¯b = eau ebu ¯ = ecv edv ¯ = ec ev ev¯ ed¯ = ec ed¯ ec =ϕ ed  ii) ϕ is linear by construction. iii) ϕ is an algebra homomorphism. First note that ϕ ee11 = e1 e¯1 = e1 . Moreover, we have ϕ  ea ec · eb ed  =ϕ  ea ec eb ed  ∗  = ea ec ed¯e¯b = ea e¯b ec ed¯ = ϕ  ea eb  ϕ  ec ed  ,  where ∗ uses the commutativity of k[M ]. iv) ϕ is a coalgebra homomorphism. Indeed, ε◦ϕ  ea eb  ∗  = ε(ea e¯b ) = ε(ea )ε(e¯b ) = ε(ea ) = ε  ea eb  ,  where ∗ uses that ε(e¯b ) = 1 (since ε(eb ) = 1). Moreover (ϕ ⊗ ϕ) ◦ ∆  ea eb  ea ea ⊗ϕ = ea e¯b ⊗ ea e¯b eb eb = ∆(ea )∆(e¯b ) = ∆(ea e¯b ) ea . =∆ ϕ eb  =ϕ  v) ϕ is an isomorphism. First note that ϕ is surjective since ϕ  ea e1  = ea and ϕ  any a ∈ M . For the injectivity, suppose that ϕ  67  ea eb  e1 ea  = ϕ  = ea¯ for ec ed  , i.e.  ea e¯b = ec ed¯. Then ea e¯b = ec ed¯  eb e¯b = e1 = ed ed¯,  and  which is equivalent to saying that  ea eb  ∼  ec ; ed  thus ϕ is injective.  In order to make this section more concrete, let us take a finite commutative monoid and compute its localization. Example 2.2.31. Consider the commutative monoid M = (Z/4Z, ·, 1) obtained by taking the factor ring (Z/4Z, +, ·) and “forgetting” its additive structure. In other words we have, by abuse of notation, Z/4Z = {0, 1, 2, 3}. Let us denote the basis vectors of the complex monoid algebra C[M ] by {e0 , e1 , e2 , e3 } and consider the monoid of group-like elements G = {e0 , e1 }. The monoid G contains zero-divisors since e0 (e1 − e2 ) = e0 − e0 = 0. Using Theorem 2.2.7 we can compute the localization C[M ][G−1 ] = C[M ] × G/ ∼. Let 3  x=  3  xi e i  and  i=0  y=  yi ei i=0  be elements of C[M ] with xi , yi ∈ C for all i. Then and only if there exist u, v ∈ G such that xu = yv  and  x g  ∼  y h  with g, h ∈ G if  gu = hv.  Since in our case G has only two elements, we can simply test the different possibilities for g, h, u and v. We then obtain that xg ∼ hy if and only if x0 + x1 + x2 + x3 = y0 + y1 + y2 + y3 . Therefore, as a complex vector space, C[M ][G−1 ] ∼ = C.  68  The next remark will give us a tool to produce new examples from known ones. Remark 2.2.32. i) Let g1 ∈ B1 and g2 ∈ B2 be group-like elements in the bialgebras B1 and B2 , respectively. Then g1 ⊗ g2 is a group-like element of H1 ⊗ H2 . ii) If (B1 , r1 ), (B2 , r2 ) are two CQT bialgebras then the bialgebra B1 ⊗ B2 has an r-form given by r : (B1 ⊗B2 )⊗(B1 ⊗B2 ) → k : (x1 ⊗x2 )⊗(y1 ⊗y2 ) → r1 (x1 ⊗y1 )r2 (x2 ⊗y2 ). iii) Let (B1 , r1 ), (B2 , r2 ) be CQT bialgebras and G1 ⊂ B1 , G2 ⊂ B2 be monoids of group-like elements. Let moreover B = B1 ⊗ B2 . Then G = {g1 ⊗g2 | g1 ∈ G1 , g2 ∈ G2 } ⊂ B is a monoid of group-like elements. Using the universal property of the localization, we find that −1 B[G−1 ] ∼ = B1 [G−1 1 ] ⊗ B2 [G2 ]  as bialgebras. Using the preceding remark, we can now build new examples. Example 2.2.33. Let H4 be Sweedler’s Hopf algebra over C (cf. Section 2.2.3) and G1 = {1, g} ⊂ H4 be a monoid of group-like elements. Let moreover C[M ] and G2 = {e0 , e1 } be as in Example 2.2.31. Then the bialgebra B = H4 ⊗ C[M ] is CQT and G = G1 ⊗ G2 is monoid of group-like elements; the localization is given by B[G−1 ] ∼ = H4 ⊗ C ∼ = H4 . Note that in this example G contains elements such as g ⊗ e0 that are not regular nor central.  69  2.3  Localization in a Coquasi-Triangular WBA  In this section we shall see how we can generalize to the weak case the localization in a CQT bialgebra presented in the previous section. We want to use again the two conditions S1 and S2 of Proposition 2.1.5 which would give us the ring structure for the localization. In the bialgebra case, it is the morphism Ig introduced in Definition 2.2.1 that enabled us to use this result. Therefore, our first task here is to check whether the “essential” properties of Ig remain true in a CQT WBA; note in particular that the definition of group-like is now more general (cf. Definition 1.1.16).  2.3.1  Construction of the Localization  We are going to construct the localization in a similar fashion to Section 2.2. Proposition 2.3.1. Let (H, r) be a CQT WBA and g ∈ G(B) a group-like element. Then the morphism Ig : H → H introduced in Definition 2.2.1 and defined by Ig (a) = r(a ⊗ g)a r−1 (a ⊗ g) is an automorphism of CQT WBA. Its inverse is given by −1 I−1 g (a) = r (a ⊗ g)a r(a ⊗ g).  Proof. In this proof we indicate by ∗ the equalities where we use that g is group-like. First note that Ig : H → H is linear by construction. The fact that I−1 g is the inverse of Ig as a map of sets results from (1.51), (1.52) and coassociativity. Let us now prove that Ig is a CQT WBA homomorphism. i) Ig preserves the unit. We have Ig (1) = r(1 ⊗ g)1 r−1 (1 ⊗ g) 70  1.19  = r(εs (1 ) ⊗ g)1 r−1 (εt (1 ) ⊗ g) ♦  = ε(gεs (1 ))1 ε(g1 ) 1.20  = ε(εs (g)εs (1 ))1 ε(εs (g)1 ) ∗  = ε(11 εs (12 ))12 ε(13 12 ) = ε(εs (1 ))1 ε(1 ) 1.19  = 1,  where ♦ follows from (1.45) and (1.48). ii) Ig preserves the product. Let a, b ∈ H, we have Ig (ab) = r((ab) ⊗ g)(ab) r−1 ((ab) ⊗ g) = r(a b ⊗ g)a b r−1 (a b ⊗ g) 1.10  = r(a b ⊗ g)a b r−1 (a ⊗ g )r−1 (b ⊗ g )  1.10  = r(b ⊗ g )r(a ⊗ g )a b r−1 (a ⊗ g )r−1 (b ⊗ g ) ♦  = r(b ⊗ g)r(a ⊗ g)ε(a b )a b ε(aIV bIV )r−1 (aV ⊗ g)r−1 (bV ⊗ g) = r(b ⊗ g)r(a ⊗ g)a b r−1 (a ⊗ g)r−1 (b ⊗ g) = r(a ⊗ g)a r−1 (a ⊗ g)r(b ⊗ g)b r−1 (b ⊗ g) = Ig (a)Ig (b), where we have used (1.53) and (1.54) for equality ♦. iii) Ig preserves the counit. Let a ∈ H, then ε(Ig (a)) = r(a ⊗ g)ε(a )r−1 (a ⊗ g) = r(a ⊗ g)r−1 (a ⊗ g) 1.52  = ε(a).  71  iv) Ig preserves the coproduct. For a ∈ H we have ∆(Ig (a)) = r(a ⊗ g)a ⊗ a r−1 (aIV ⊗ g) = r(a ⊗ g)a ε(a ) ⊗ aIV r−1 (aV ⊗ g) 1.51  = r(a ⊗ g)a r−1 (a ⊗ g) ⊗ r(aIV ⊗ g)aV r−1 (aV I ⊗ g)  = (Ig ⊗ Ig )∆(a). v) Ig preserves the r-form. For a, b ∈ H we have r ◦ (Ig (a) ⊗ Ig (b)) = r(r(a ⊗ g)a r−1 (a ⊗ g) ⊗ r(b ⊗ g)b r−1 (b ⊗ g)) = r(a ⊗ g)r(a ⊗ b )r−1 (a ⊗ g)r(b ⊗ g)r−1 (b ⊗ g) 1.11  = r(a ⊗ gb )r−1 (a ⊗ g)r(b ⊗ g)r−1 (b ⊗ g) = r(a ⊗ gr(b ⊗ g)b r−1 (b ⊗ g))r−1 (a ⊗ g) = r(a ⊗ gIg (b))r−1 (a ⊗ g)  2.26  = r(a ⊗ bg)r−1 (a ⊗ g)  1.11  = r(a ⊗ b)r(a ⊗ g)r−1 (a ⊗ g)  1.52  = r(a ⊗ b)ε(a ) = r(a ⊗ b).  Proposition 2.3.2. Let (H, r) be a CQT WBA and G ⊂ G(H) a monoid of group-like elements. Then I : G → Aut(H) : g → Ig is an anti-morphism of monoids.  72  Proof. Let a ∈ H. Let us first check that I preserves the unit. Indeed 1.28  I1 (a) = r(a ⊗ 1)a r−1 (a ⊗ 1) = ε(a )a ε(a ) = id(a). Let us now verify that Igh = Ih ◦ Ig , i.e. that I “anti-preserves” multiplication. Let g, h ∈ G(H), then Ih (Ig (a)) = r(a ⊗ g)r(a ⊗ h)a r−1 (aIV ⊗ h)r−1 (aV ⊗ g) 1.11  = r(a ⊗ gh)a r−1 (a ⊗ gh) = Igh (a).  The next proposition will enable us to prove that conditions S1 and S2 of Prop. 2.1.5 are satisfied by (H, r) and G ⊂ G(H) a monoid of group-like elements. Proposition 2.3.3. Let (H, r) be a CQT WBA, a ∈ H and g ∈ G(H) a group-like element. Then ag = gIg (a). (2.26) Proof. We have gIg (a) = r(a ⊗ g)ga r−1 (a ⊗ g) = r(a ⊗ g)g(1a )r−1 (a ⊗ g) = r(a ⊗ g)ε(1 a )g1 a r−1 (aIV ⊗ g) 1.44  = r(a ⊗ g)r(a ⊗ εs (1 ))g1 a r−1 (aIV ⊗ g)  1.19  = r(a ⊗ g)r(a ⊗ 1 )g1 a r−1 (aIV ⊗ g)  1.11  = r(a ⊗ g1 )g1 a r−1 (a ⊗ g) ∗  = r(a ⊗ g )g a r−1 (a ⊗ g) 1.9  = a g r(a ⊗ g )r−1 (a ⊗ g) 73  ∗  = a 1 gr(a ⊗ 1 g)r−1 (a ⊗ g) 1.11  = a 1 gr(a ⊗ 1 )r(a ⊗ g)r−1 (aIV ⊗ g)  1.19  = a 1 gr(a ⊗ εt (1 ))r(a ⊗ g)r−1 (aIV ⊗ g)  1.43  = (a1) gε((a1) )r(a ⊗ g)r−1 (aIV ⊗ g)  1.52  = a gε(a )  = ag, where ∗ holds because g is group-like. Lemma 2.3.4. Let (H, r) be a CQT WBA and G a monoid of group-like elements such that Ig (G) ⊂ G for all g ∈ G. Then conditions S1 and S2 of Prop. 2.1.5 hold. Proof. S1 : for a ∈ H, g ∈ G, we have gIg (a) = ag by (2.26), S2 : let a ∈ H, g ∈ G be such that ga = 0. Then 2.26  −1 0 = Ig (ga) = Ig (I−1 g (a)g) = Ig (Ig (a))Ig (g) = aIg (g).  Thanks to these results we can now use Proposition 2.1.5 and we get the following Theorem 2.3.5. Let (H, r) be a CQT WBA and G be a monoid of group-like elements such that Ig (G) ⊂ G for all g ∈ G. Define the equivalence relation ∼ on H × G as follows : (a1 , g1 ) ∼ (a2 , g2 ) if there exist u1 , u2 ∈ H such that a1 u1 = a2 u2 ∈ H g1 u1 = g2 u2 ∈ G.  and  (2.27) (2.28)  Let us denote by ag the equivalence class containing (a, g). Then H[G−1 ] := (H × G)/ ∼ has a weak bialgebra structure. Its unital algebra structure is 74  given by a + g a · g  b aIg (h) + bg = , h hg b aIg (b) = , h hg  where 11 is the unit and scalar multiplication is given by α · ag = and its counital coalgebra structure is given by ∆  a g  =  a a ⊗ g g  and  ε  a g  αa g  for α ∈ k  = εH (a).  Before proving this theorem let us note that Lemma 2.2.8, which proves that ∼ is an equivalence relation, still holds in the weak case. Concerning Lemma 2.2.9, which states that u1 and u2 of (2.27) can be taken in G, the result is still valid (as we shall see in the next lemma) even though we cannot simply use the same proof. Indeed, in the weak case we don’t know whether the morphism Ig is the identity on group-like elements or not. Lemma 2.3.6. Let (H, r) and G be as in Thm. 2.3.5. Then ag11 ∼ ag22 if and only if there exist v1 , v2 ∈ G such that a1 v1 = a2 v2 and g1 v1 = g2 v2 ∈ G. Proof. Obviously, if there exist v1 , v2 ∈ G such that a1 v1 = a2 v2 and g1 v1 = g2 v2 ∈ G then ag11 ∼ ag22 . Now suppose that ag11 ∼ ag22 ; we have to show that we can find v1 , v2 in G (and not simply in H) such that a1 v 1 = a2 v 2  and g1 v1 = g2 v2 ∈ G.  By definition there exist u1 , u2 ∈ H such that a1 u1 = a2 u2  and g1 u1 = g2 u2 ∈ G.  2.26  Then, g1 Ig1 (g1 u1 ) = g1 u1 g1 and hence by S2’ there exist w1 ∈ G such that 75  Ig1 (g1 u1 )w1 = u1 g1 w1 , which belongs to G since both Ig1 (g1 u1 ) and w1 do. 2.26 Similarly g2 Ig2 (g2 u2 ) = g2 u2 g2 implies the existence of w2 ∈ G such that Ig2 (g2 u2 )w2 = u2 g2 w2 ∈ G. Therefore both v1 := u1 g1 w1 Ig1 w1 (g2 w2 ) and v2 := u2 g2 w2 g1 w1 are in G. Then 2.26  a1 v1 = a1 u1 g1 w1 Ig1 w1 (g2 w2 ) = a2 u2 g1 w1 Ig1 w1 (g2 w2 ) = a2 u2 g2 w2 g1 w1 = a2 v2 and 2.26  g1 v1 = g1 u1 g1 w1 Ig1 w1 (g2 w2 ) = g2 u2 g1 w1 Ig1 w1 (g2 w2 ) = g2 u2 g2 w2 g1 w1 = g2 v2 .  Notation 2.3.7. In analogy to the bialgebra case, we denote by ρ the homomorphism a ρ : H → H[G−1 ] : a → . 1 Proof of Thm. 2.3.5. i) One proves in a similar way to the bialgebra case −1 that H[G ] is a ring. ii) H[G−1 ] is a k-vector space. We have, for α, β ∈ k, b αa αb (αa)Ig (h) + (αb)g a + = α +α = g h g h hg aIg (h) bg aIg (h) bg =α + =α + hg hg gIg (h) hg a b + . =α g h Moreover 1·  a 1·a a = = , g g g  (αβ)a α(βa) βa a a (αβ) = = =α =α β g g g g g 76  ,  and (α + β)a αa + βa αa βa a a a = = + =α +β . (α + β) = g g g g g g g iii) H[G−1 ] is a unital algebra. The only verification that needs to be done is, for α ∈ k, α  a g  b a = h g  α  b h  =α  a b · g h  .  And indeed, α  αa b αaIg (b) aIg (αb) a αb a b = · = = = · = h g h hg hg g h g  a g  α  where we have used the linearity of Ig . Moreover, α  a g  αa b αaIg (b) aIg (b) b = · = =α =α h g h hg hg  a b · g h  iv) The comultiplication is well-defined. Let ag11 ∼ ag22 , i.e. there exist u1 , u2 ∈ G such that a1 u1 = a2 u2 and g1 u1 = g2 u2 . Then ∆  a1 g1  a1 a1 a u1 a u1 ⊗ = 1 ⊗ 1 g1 g1 g1 u1 g1 u1 a1 u1 a1 u1 1 1 ⊗ · ⊗ = 1 1 g1 u1 g1 u1 1 a2 u2 a2 u2 1 ♦ = ⊗ · ⊗ 1 1 g2 u2 g2 u2 =  77  .  b h  a2 u2 a2 u2 a a ⊗ = 2⊗ 2 g2 u2 g2 u2 g2 g2 a2 , =∆ g2 =  where equality ♦ holds because for g group-like we have ∆(ag) = a g ⊗ a g by (1.50) and thus a1 u1 ⊗ a1 u1 = ∆(a1 u1 ) = ∆(a2 u2 ) = a2 u2 ⊗ a2 u2 . We then apply ρ ⊗ ρ to this last equality. v) The comultiplication is k-linear. For α, β ∈ k, we have ∆ α  a g  αa αa a = ⊗ g g g a a a a =α ⊗ =α ⊗ g g g g a = α∆ . g =∆  Moreover ∆  a b + g h  aIg (h) + bg hg (aIg (h) + bg) (aIg (h) + bg) = ⊗ hg hg (aIg (h)) (bg) (bg) ∗ (aIg (h)) ⊗ + ⊗ = hg hg hg hg a Ig (h) b g b g ♦ a Ig (h) ⊗ + ⊗ = gIg (h) gIg (h) hg hg a a b b = ⊗ + ⊗ g g h h a b =∆ +∆ g h =∆  where ∗ follows from the linearity of the comultiplication in H and ♦ from the fact that for a group-like g, we have ∆(ag) = a g ⊗ a g by 78  (1.50) and ag = gIg (a) by (2.26). vi) The counit is well-defined. Take ag11 ∼ ag22 as before. Then ε  a1 g1  1.49  1.49  = ε(a1 ) = ε(a1 u1 ) = ε(a2 u2 ) = ε(a2 ) = ε  a2 g2  .  vii) The counit is k-linear. Indeed, we have ε α  a g  =ε  αa g  = ε(αa) = αε(a) = αε  a g  ,  and ε  a b + g h  aIg (h) + bg = ε(aIg (h) + bg) hg = ε(aIg (h)) + ε(bg) = ε(aεt (Ig (h))) + ε(bεt (g))  =ε  = ε(a · 1) + ε(b · 1) = ε(a) + ε(b) a b =ε +ε , g h where we have used that Ig (h) and g are group-like. viii) (H[G−1 ], ∆, ε) is a counital coalgebra. We have (ε ⊗ id) ◦ ∆  a g  =ε  (id ⊗ ε) ◦ ∆  a g  =  a g  a a ε(a )a a = ε(a ) = = , g g g g  and a ε g  a g  79  = ε(a )  a ε(a )a a = = . g g g  Moreover, noting that (a ) ⊗ (a ) ⊗ a = a ⊗ (a ) ⊗ (a ) becomes (a ) a a a a a (a ) (a ) (a ) ⊗ ⊗ = ⊗ ⊗ = ⊗ ⊗ 1 1 1 1 1 1 1 1 1 once we apply ρ ⊗ ρ ⊗ ρ to it, we find (∆ ⊗ id) ◦ ∆  a g  a a (a ) (a ) a ⊗ = ⊗ ⊗ g g g g g 1 1 1 (a ) (a ) a = ⊗ ⊗ ⊗ ⊗ 1 1 1 g g g a (a ) (a ) 1 1 1 = ⊗ ⊗ ⊗ ⊗ 1 1 1 g g g a (a ) (a ) a a = ⊗ ⊗ = ⊗∆ g g g g g a . = (id ⊗ ∆) ◦ ∆ g =∆  ix) Multiplicativity of the coproduct. We have ∆  a b · g h  aIg (b) (aIg (b)) (aIg (b)) = ⊗ hg hg hg a Ig (b) a Ig (b) a b a b = ⊗ = · ⊗ · hg hg g h g h a a b b = ⊗ · ⊗ g g h h a b =∆ ∆ . g h =∆  x) Weak comultiplicativity of the unit. Applying ρ ⊗ ρ ⊗ ρ to 1 ⊗ 1 ⊗ 1 = 11 ⊗ 11 12 ⊗ 12 we get 1 1 1 1 1 1 1 ⊗ ⊗ = 1 ⊗ 1 2 ⊗ 2. 1 1 1 1 1 1 80  Then we have 1 1 1 1 1 1 1.28 1 1 I1 (12 ) 12 1 ⊗ ⊗ = 1⊗ 1 2⊗ 2 = 1⊗ 1 ⊗ 1 1 1 1 1 1 1 1 1 11 11 12 12 ⊗ · ⊗ . = 1 1 1 1 Similarly we find that 1 1 1 1 1 1 1 ⊗ ⊗ = 1 ⊗ 2 · 1 ⊗ 2. 1 1 1 1 1 1 1 xi) Weak multiplicativity of the unit. We have ε  a b c · · g h k  aIg (b) c aIg (b)Ihg (c) · =ε hg k khg = ε(aIg (b)Ihg (c)) = ε(aIg (b) )ε(Ig (b) Ihg (c)) =ε  = ε(aIg (b ))ε(Ig (b )Ihg (c)) = ε(aIg (b ))ε(Ig (b Ih (c))) aIg (b ) b Ih (c) ∗ = ε(aIg (b ))ε(b Ih (c)) = ε ε hg kh b c a b · ε · , =ε g h h k where ∗ holds since Ig preserves the counit of H by Proposition 2.3.1. In a similar fashion, we find that ε  a b c · · g h k  =ε  a b · g h  ε  b c · h k  .  Proposition 2.3.8. Let (H, r) and G ⊂ G(H) be as above. Then ρ : H → H[G−1 ] : a → a1 is a WBA homomorphism and H has the following ¯ such that universal property: for any WBA homomorphism ψ : H → H ψ(g) is invertible for any g ∈ G, there exists a unique WBA homomorphism 81  ¯ such that ψ¯ : H[G−1 ] → H H  ρ  /  H[G−1 ] #   ψ  ψ¯  ¯ H.  commutes. The proof follows directly from the bialgebra case (cf. Proposition 2.2.14). Remark 2.3.9. Note that the homomorphism ρ : H → H[G−1 ] of the preceding proposition satisfies the conditions of Definition 2.1.2. In other words, H[G−1 ] is the ring of fractions of H with respect to G (on which we have built a WBA structure). Corollary 2.3.10. When the localization exists, it is unique up to a unique isomorphism.  2.3.2  Example : Monoid Generated by a Central Grouplike Element  In this section we shall see that a similar phenomenon to the one observed in Section 2.2.2 also holds in the weak case; more precisely, if H is a WBA and the monoid G is generated by a central group-like element g, then the localization has the form H[t]/(gt − 1). Proposition 2.3.11. Let H be a WBA and let g be a central group-like element. Assume that G is the monoid freely generated by g, i.e. G = {1, g, g 2 , . . . }. Then the localization is given by H[G−1 ] ∼ = H[t]/(gt − 1). Proof. We know that the localization H[G−1 ] exists and that in this case it 82  has a weak bialgebra structure given by, for a1 , a2 ∈ H and g1 , g2 ∈ G, a2 a1 g2 + a2 g1 = g2 g1 g2 a2 a1 a2 = g2 g1 g2 1 is the unit, 1 a1 a a = 1 ⊗ 1, g1 g1 g1  a1 + g1 a1 · g1  ∆  since g1 g2 = g2 g1 , since g1 a2 = a2 g1 and g1 g2 = g2 g1 ,  a1 g1  and ε  = εH (a1 ).  First, let us show that as an algebra, the localization of H at G is given by H[t]/(gt − 1). We do this by proving that H[t]/(gt − 1) satisfies the universal property of the localization. Let ψ : H → H be an algebra homomorphism with ψ(g) invertible for any g ∈ G. Let us check that there exists a unique algebra homomorphism ψ¯ : H[t]/(gt − 1) → H such that H  i  /  H[t]/(gt − 1) ψ¯  ψ  #   H  commutes, with i : H → H[t]/(gt − 1) given by i(a) = a. ¯ Since we want ψ¯ ◦ i = ψ, we are forced to define ψ¯ by ψ(a) := ψ(a) for any ¯ := ψ(g)−1 in a ∈ H. Moreover, t being the inverse of g, we have to set ψ(t) order for ψ¯ to be well-defined. In other words, ψ¯ is completely determined by ψ. Furthermore, ψ¯ is indeed well-defined since ¯ − 1) = ψ(g) ¯ ψ(t) ¯ − ψ(1) ¯ ψ(gt = ψ(g)ψ(g)−1 − 1 = 0. As a second step, we use this algebra isomorphism to define a weak bialgebra structure on H[t]/(gt − 1). Using the above universal property for H[t]/(gt − 1) with i : H → H[t]/(gt − 1) : a → a and H[G−1 ] with ρ : H → H[G−1 ] :  83  a → a1 , we get the following commutative diagram H[t]/(gt − 1) 9  i ρ  H  i  /  ρ¯    H[G−1 ] %    ¯i  H[t]/(gt − 1) with ρ¯ : H[t]/(gt − 1) → H[G−1 ] : atn →  a gn  and  ¯i : H[G−1 ] → H[t]/(gt − 1) : a → atn . gn By the universal property ¯i and ρ¯ are inverse of each other and thus algebra isomorphisms. Using that ¯i is moreover a coalgebra homomorphism, we then define the coalgebra structure of H[t]/(gt − 1) by a gn = a tn ⊗ a tn ,  ∆(atn ) = ∆ ¯i  = (¯i ⊗ ¯i) ◦ ∆  a gn  = (¯i ⊗ ¯i)  a a ⊗ n n g g  for the comultiplication whereas for the counit we have ε(atn ) = ε ¯i and ε(tn ) = ε ¯i  Remark 2.3.12.  a gn 1 gn  =ε  =ε  a gn 1 gn  = ε(a)  = εH (1).  i) First, note that since H is a WBA, εH (1) = 1 in 84  general. ii) Second, this construction can be seen in the following way. Since we know that H[G−1 ] ∼ = H[t]/(gt − 1) as algebras, we then define the coalgebra structure on H[t]/(gt − 1) such that the morphism ρ¯ becomes a coalgebra homomorphism as well and thus H[G−1 ] becomes isomorphic to H[t]/(gt − 1) as WBAs. iii) Finally, note that this construction can be extended to the case where G is generated by finitely many central group-like elements. Remark 2.3.13. This example illustrates that if the monoid G is in the centre of H, then the localization H[G−1 ] exists even if H is not coquasitriangular. Indeed, the centrality of G is enough for conditions S1 and S2 of Proposition 2.1.5 to be satisfied and then for the localization to exist.  2.3.3  Example : Weak Bialgebra Associated with a Finite Directed Graph  In this section we use the theory developed in [Pfe11] to construct the weak bialgebra associated with a finite graph and then localize it. As we focus here on the localization, the reader interested in knowing more about this type of weak bialgebra is referred to Sections 3 and 6 of the above article. Construction 2.3.14. Let G be a finite directed graph whose set of vertices is denoted by G 0 and the set of edges by G 1 ⊆ G 0 × G 0 . Every edge p = (v0 , v1 ) ∈ G 1 has a source and a target vertex, denoted respectively by σ(p) = v1 ∈ G 0 and τ (p) = v0 ∈ G 0 . We also set σ(v) = v = τ (v) for all v ∈ G 0 . The paths of length m, for any m ∈ N, are denoted by m  G m = { (p1 , . . . , pm ) ∈ (G 1 ) | σ(pj ) = τ (pj+1 ) for all 1  j  m − 1 }.  The concatenation of two paths p ∈ G l and q ∈ G m with σ(p) = τ (q) is 85  denoted pq ∈ G l+m . We write CG m , with m ∈ N0 , for the free C-vector space on the set G m . The vector space (CG m )∗ ⊗ CG m  H[G] = m∈N0  has then a WBA structure given by µ([p|q]m ⊗ [r|s]l ) = δσ(p),τ (r) δσ(q),τ (s) [pr|qs]m+l , η(1) =  [j|l]0 , j,l∈G 0  [p|r]m ⊗ [r|q]m ,  ∆([p|q]m ) = r∈G m  ε([p|q]m ) = δpq , for all p, q ∈ G m , r, s ∈ G l , m, l ∈ N0 . Note that the coproduct is taken in the category VectC of C-vector spaces and that we write [p|q]m for the homogeneous element p ⊗ q ∈ (CG m )∗ ⊗ CG m . As usual, we write δpq = 1 if p = q and δpq = 0 if p = q for all p, q ∈ G m , m ∈ N0 . Observe that as an algebra, H[G] ∼ = k(G × G) is the path algebra of the directed graph G × G, and thus is graded with respect to path length m ∈ N0 . As a coalgebra, it is a direct sum of matrix coalgebras; one for each path length. Now that we have constructed a weak bialgebra structure on H[G], let us look at its localization. Example 2.3.15. Let us consider the graph G given by )1  0  •h  •.  Since in this graph, the two vertices are connected by one edge “in each direction”, we can denote a path p ∈ G m of length m by a sequence of m + 1 86  vertices, namely, p = (i0 , . . . , im ) ∈ (G 0 )m+1 . The source of this path is σ(p) = im and its target τ (p) = i0 . Let H[G] be the WBA constructed for the graph G above. The multiplication of two elements consist simply in the concatenation of the paths in each component (and is zero if the paths can not be concatenated). The unit is η(1) = [(0)|(0)]0 + [(0)|(1)]0 + [(1)|(0)]0 + [(1)|(1)]0 and the counit is 1 if the two paths are equal and zero otherwise. Consider the element of g ∈ H[G] given by g = [(0, 1, 0)|(0, 1, 0)]2 + [(1, 0, 1)|(1, 0, 1)]2 −[(0, 1, 0)|(1, 0, 1)]2 − [(1, 0, 1)|(0, 1, 0)]2 . One checks by direct computation that g is both central and group-like in H[G]. Let G be the monoid generated by g. Since G is in the centre of H[G], we know that we can localize H[G] relative to G. Of course, one can use the structure of the localization given by Theorem 2.3.5 and see elements of H[G][G−1 ] as fractions. In the present case though, it may be simpler to apply Proposition 2.3.11 and consider the localization as the polynomial algebra H[G][t]/(gt − 1). This illustrates how weak bialgebra of fractions can sometimes “fold” into a polynomial algebra.  87  Chapter 3 Hopf Envelope In this chapter we first follow the works of Takeuchi [Tak71], Manin [Man88] and Pareigis [Par] and construct the so-called Hopf envelope of a bialgebra. We then define the more general notion of weak Hopf envelope and briefly discuss its relationship with the localization.  3.1  Manin’s Hopf Envelope  In [Man88, Chap. 7] Manin studies specific quotients of matrix bialgebras and builds a Hopf algebra H(B) having the following universal property. Let B be a certain quotient of a matrix-bialgebra. There exists a Hopf algebra H(B) and a bialgebra homomorphism i : B → H(B) such that for any Hopf algebra H and any bialgebra homomorphism f : B → H there exists a unique Hopf algebra homomorphism f : H(B) → H satisfying f = f ◦ i, i.e. such that i / B H(B) "   f  f  H  commutes. To construct such a Hopf algebra, Manin uses a technique similar to the one developed by Takeuchi in [Tak71]. 88  We are first going to give this result as presented in [Man88, Chap. 7]. In the next section, we shall then give an alternative formulation in a slightly more general context and exhibit some examples. Let Z be a family of symbols (tij )1 i,j N . Let k be a field and A be the matrix bialgebra generated by Z, i.e. the free algebra over Z with coefficients in k and comultiplication given by tim ⊗ tmj  ∆(tij ) = 1 m N  and counit by ε(tij ) = δij . Let R = {Cijmn | 1  i, j, m, n  N } ⊂ A be a set with the Cijmn ’s defined by cqr ij tqm trn −  Cijmn = 1 q,r N  tiq tjr cmn qr , 1 q,r N  where the cmn ij ’s are elements of the ground field k. We can then take the quotient of A by (R), the two-sided ideal generated by the Cijmn ’s. It turns out that this ideal is also a coideal and therefore the quotient is a bialgebra. Let B := A/(R). Note that the construction of such a quotient of a matrix bialgebra is nowadays referred as the “FRT construction” because it was studied systematically by Faddeev, Reshetikhin and Takhtadjian in [RTF89]. 1 (1)  Define a series of families of symbols by Z0 = Z, Z1 = {tij | 1 i, j (2) N }, Z2 = {tij | 1 i, j N }, . . . . Let H = k{Z0 , Z1 , . . . } be the free algebra generated by Z0 , Z1 , . . . and let H(B) be H quotiented by the ideal generated by the following relations (where all the sums are taken over m 1  Another good introduction on the topic is [Kas95, Chap. VIII.6].  89  running from 1 to N ) : Rn =  elements of R written for Zn if n ≡ 0 op,cop elements of R written for Zn if n ≡ 1  (n) (n+1)  tim tmj  − 1,  (3.1)  (n+1) (n) tmj  − 1,  for n ≡ 0  mod 2  (3.2)  (n+1) (n) tjm  − 1,  for n ≡ 1  mod 2.  (3.3)  tim  m  mod 2 mod 2  m (n) (n+1)  tmi tjm  − 1,  tmi  m  m  Define i : B → H(B) as the inclusion of B in “shift” 0, i.e. i(B) = Z0 /(R). Note that here the superscripts over the t’s indicates which family the symbol belongs to and is not a power. The next theorem ensures that this construction makes sense and that H(B) has the desired universal property. Theorem 3.1.1 ([Man88, Thm. 3, Chap. 7]). a) Relations (3.1) - (3.3) generate a coideal R with respect to the comultiplication ∆ : H → H⊗H given by (n) ∆(tij )  =  (n) m tim (n) m tmj  (n)  ⊗ tmj for n ≡ 0 (n) ⊗ tim for n ≡ 1  mod 2 mod 2  and thus ∆ induces a comultiplication ∆ : H(B) → H(B) ⊗ H(B). (n) Together with ε(tij ) = 1 this makes H(B) a bialgebra and i : B → H(B) a bialgebra homomorphism. (n)  (n+1)  b) The map S : H → H, defined by S(tij ) = tij and S(ab) = S(b)S(a) for any a, b ∈ H, satisfies S(R) ⊂ R. It thus induces a linear map S : H(B) → H(B) which is the antipode of H(B). c) For any bialgebra homomorphism f : B → H, where H is a Hopf algebra, there exists a unique Hopf algebra homomorphism f : H(B) →  90  H such that B  i  f  /  H(B) "   f  H  commutes. We shall explain (and reword in a more general context) this construction in detail in Section 3.2.1. Nevertheless, let us summarise the main steps of this procedure right now. Consider the bialgebra B. The first step is is to form the coproduct B := B B op,cop B . . . in the category of bialgebras. This corresponds to constructing H and quotienting by (3.1). (n+1) (n) Second, define the“pre-antipode” S : B → B : tij → tij . In other words, (n) S increases the “shift” of tij by one. This is part b) in Theorem 3.1.1 Third, form the ideal generated by the relations {(µ ◦ (S ⊗ idB ) ◦ ∆ − η ◦ ε)(x), (µ ◦ (idB ⊗ S) ◦ ∆ − η ◦ ε)(x) | x ∈ B}, and then quotient by this ideal. This corresponds to dividing by relations (3.2) and (3.3) in Manin’s work. Part a) of the previous theorem assures that it is a coideal as well and part c) that the newly formed H(B) has the desired universal property.  3.2  Reformulation and Examples  Our goal is to have a description of the Hopf envelope that can be used for any bialgebra. To this end we reformulate Manin’s result using categorical language. We shall then construct examples of Hopf envelopes.  91  3.2.1  Categorical Reformulation of Manin’s Theorem  In order to be able to use Manin’s result in more general situations, we describe this construction in categorical language. We get the following slightly more general Theorem 3.2.1. Let B be a bialgebra. Then there exists a Hopf algebra H(B) and a bialgebra homomorphism i : B → H(B) satisfying the following universal property : for any Hopf algebra H and bialgebra homomorphism f : B → H, there exists a unique Hopf algebra homomorphism f : H(B) → H making the following diagram commute B  i  /  H(B) "   f  f  H.  Definition 3.2.2. The Hopf algebra H(B) of the theorem is called the Hopf envelope of B. Proof. Define a series of bialgebras by B0 := B and Bn+1 := Bnop,cop , n ∈ N0 , hence when n is even Bn is simply B and when n is odd Bn is the opposite-coopposite bialgebra of B (i.e. B with the opposite multiplication and opposite comultiplication). Then, let B := n∈N0 Bn be the coproduct of the Bn ’s with injections in : Bn → B. Using the universal property of the coproduct, we know that op,cop there exists a unique bialgebra homomorphism S : B → B making the diagram in /B Bn (3.4) id    op,cop Bn+1  in+1  commute for any n ∈ N0 . 92  /    B  S  op,cop  Let I be the two-sided ideal in B generated by {(µ ◦ (S ⊗ idB ) ◦ ∆ − η ◦ ε)(x), (µ ◦ (idB ⊗ S) ◦ ∆ − η ◦ ε)(x) | x ∈ B}. Remark 3.2.3. Note that here it is not possible to use exclusively the categorical language and define the quotient by I as a coequaliser in the category of bialgebras since the morphism µ ◦ (S ⊗ idB ) ◦ ∆ defining I is not an algebra homomorphism but only a linear map. Notation 3.2.4. In order to make the subsequent computations easy to read, we use the following notations. We write εn := εBn , ∆n := ∆Bn and in general, the subscript n indicates Bn and no subscript indicates B (e.g. ∆ = ∆B ). Moreover, the subscript k indicate that we are in the field, e.g. µk is the multiplication in the field k.  Let us now return to the proof. In order to take the quotient by I and get a bialgebra, we first have to show that I is also a two-sided coideal, i.e. I ⊂ ker εB and ∆B (I) ⊂ I ⊗ B + B ⊗ I. It is enough to check these conditions on generating elements of I, i.e. elements y ∈ I of the form y = in (x) for some x ∈ Bn , n ∈ N0 . So, let x be in some Bn . Then, ε(µ(S ⊗ idB )∆in (x)) = µk (εS ⊗ ε)(in ⊗ in )∆n (x) 3.4  = µk (εin+1 idBn ⊗ εin )∆n (x)  = εin+1 idBn (x) = εn+1 (x) = εn (x) = ε(ηεin (x)), where  op,cop holds since Bn = (B, µ, η, ∆, ε) = Bn+1 if n is even and Bn =  93  op,cop (B, µop , η, ∆op , ε) = Bn+1 if n is odd. Similarly we have ε(µ(idB ⊗S)∆in (x)) = ε(ηεin (x)). Moreover, using Sweedler’s notation and σ(v ⊗ w) := w ⊗ v for the map flipping the two tensor factors, we have  ∆(µ(S ⊗ id)∆in (x)) = (µ ⊗ µ)(id ⊗ σ ⊗ id)(∆ ⊗ ∆)(S ⊗ id)(in ⊗ in )∆n (x) = (µ ⊗ µ)(id ⊗ σ ⊗ id)(σ(S ⊗ S)∆ ⊗ ∆)(in ⊗ in )∆n (x) = (µ ⊗ µ)(id ⊗ σ ⊗ id)(Sin (x ) ⊗ Sin (x ) ⊗ in (x ) ⊗ in (xIV )) = Sin (x )in (x ) ⊗ Sin (x )in (xIV ) = (S ∗ id)(x ) ⊗ Sin (x )in (x ), where (S ∗ id) is the convolution product of S with id. Then adding the −ηεin (x) term, we get ∆(µ(S ⊗ id)∆in (x) − ηεin (x)) = (S ∗ id)(x ) ⊗ Sin (x )in (x ) − ∆ηεin (x) = ((S ∗ id) − ηε)(x ) ⊗ Sin (x )in (x ) + (ηε)(x ) ⊗ Sin (x )in (x ) − ∆ηεin (x) = ((S ∗ id) − ηε)(x ) ⊗ Sin (x )in (x ) + η(1) ⊗ Sin (x )in (x ) − η(1) ⊗ ηεin (x) = ((S ∗ id) − ηε)(x ) ⊗ Sin (x )in (x ) + η(1) ⊗ (S ∗ id − ηε)in (x) ∈ I ⊗ B + B ⊗ I, hence I is a two-sided coideal. Since I is also an ideal by definition, the quotient H(B) = H/I is a bialgebra and the canonical projection π : B → H/I is a bialgebra homomorphism. Next we show that H(B) is actually a Hopf algebra with antipode induced 94  by S. To this end we first have to verify that S(I) ⊂ I. On the one hand, we have S((S ∗ id)in (x)) = S(µ(S ⊗ id)∆in (x)) 2  = µσ(S ⊗ S)(in ⊗ in )∆n (x) 3.4  = µσ(S ⊗ id)(in+1 ⊗ in+1 )∆n (x)  = µ(id ⊗ S)(in+1 ⊗ in+1 )σ∆n (x) = µ(id ⊗ S)(in+1 ⊗ in+1 )∆n+1 (x) = (id ∗ S)in+1 (x) and on the other hand S(ηεin (x)) = S(η(1))εn (x) = S(η(1))εn+1 (x) = S(ηεin+1 (x)). Hence S((S ∗ id − ηε)in (x)) = (id ∗ S − ηε)in+1 (x) ∈ I. Together with the symmetric statement, this shows that S(I) ⊂ I. Then, since ker π = I, we have that I ⊂ ker(π ◦ S). Therefore the morphism π ◦ S : B → H(B) factors through H(B) and there is a unique morphism S : H(B) → H(B) such that S ◦ π = π ◦ S, or in terms of diagrams, such that π / H(B) B (3.5) S  B    op,cop π  /    S  H(B)op,cop  commutes. Since S descends to the quotient, and that the ideal I by which we mod out is generated precisely by the relations making S the antipode, it is straightforward that S : H(B) → H(B)op,cop is the antipode of H(B), in other words, that H(B) is a Hopf algebra. 95  We finally have to prove that H(B) has the universal property stated above. Let H be a Hopf algebra and f : B → H a bialgebra homomorphism. We have to show that there exists a unique Hopf algebra homomorphism f : H(B) → H making the following diagram commute B  i  /  H(B) "   f  f  H,  where i : H → H(B) is the composite i = π ◦ i0 . n Define the bialgebra homomorphisms fn : Bn → H by fn = SH f for all n ∈ N0 ; in particular, f0 = f . Then, by the universal property of the coproduct, there exists a unique bialgebra homomorphism f : B = Bn → H such that f in = fn for all n ∈ N0 . We now show that I ⊂ ker f and therefore f factors through the quotient H(B). Let x ∈ Bn , then f ((S ∗ id)in (x)) = f µ(S ⊗ id)∆in (x) = f µ(S ⊗ id)(in ⊗ in )∆n (x) = µ(f Sin ⊗ f in )∆n (x) 3.4  = µ(f in+1 ⊗ f in )∆n (x)  = µ(fn+1 ⊗ fn )∆n (x) = µ(SH fn ⊗ fn )∆n (x) = µ(S ⊗ id)∆n fn (x) = (S ∗ id)fn (x) = ηεf in (x) = f (ηεin (x)). Together with the symmetric statement, this proves that I ⊂ ker f . Thus  96  there exists a unique bialgebra homomorphism f : H(B) → H such that f = f π. Hence we get the following commutative diagram B = B0  i0  /  B=  B  /  π  H(B) = B/I  f f  f  $)   H.  Since every bialgebra homomorphism between Hopf algebras preserves the antipode, we know that f is indeed a Hopf algebra homomorphism. Remark 3.2.5. At the beginning of the proof on page 92 we took the index “n” to be in N0 and we constructed a Hopf algebra. Note that if we want to solve this universal problem in the full subcategory of Hopf algebra with bijective antipodes we then need to take n ∈ Z. Indeed, since the antipode is given by S(x(n) ) = x(n+1) , if we take n ∈ Z we can then construct the inverse of the antipode by defining S −1 (x(n) ) = x(n−1) . In other words, the antipode is bijective. Similarly, if one is interested in the full subcategory of Hopf algebras with antipode of order 2m, then one takes n ∈ Z2m since this forces the order of the antipode to be 2m.  3.2.2  Example : Hopf Envelope of Mq (2)  Let us consider the bialgebra Mq (2) introduced in Section 1.2.1; how does its Hopf envelope H(Mq (2)) look like? Following the procedure described in the preceding section, we have the following steps. 1. Define a series of bialgebras by Bn = Mq (2) if n is even and Bn = Mq (2)op,cop if n is odd. Then form the bialgebra B as the coproduct of these bialgebras, i.e. B= Bn . n∈N0  97  2. Using the universal property of the coproduct, we define the unique op,cop bialgebra homomorphism S : B → B making the diagram in  Bn  /  B  id    op,cop Bn+1  in+1  /    B  S  op,cop  (n)  (n+1)  commute. Explicitly we have S(tij ) = tij , where the superscript (n) indicates the “shift” inside B in which tij is (and is not a power). 3. Let us now define the ideal I. It is generated by {(µ ◦ (S ⊗ idB ) ◦ ∆ − η ◦ ε)(x), (µ ◦ (idB ⊗ S) ◦ ∆ − η ◦ ε)(x) | x ∈ B}. On the generators of B this becomes µ ◦ (S ⊗ id) ◦ ∆  t11 t12 t21 t22   = µ ◦ (S ⊗ id)   = µ  =  t11 t12 t21 t22  t11 t12 t21 t22  t11 t12 t21 t22 (n+1)  ⊗  (n+1)  (n)  = (n)     (n)  t11 t12 t21 t22  t11 t12 t21 t22  ·  t11 t12 t21 t22   ⊗  (n)    (n)  ,  and similarly  µ ◦ (id ⊗ S) ◦ ∆  t11 t12 t21 t22  (n)  =  98  t11 t12 t21 t22  (n)  ·  t11 t12 t21 t22  (n+1)  .  We moreover have  η ◦ ε  t11 t12 t21 t22  (n)   =  1 0 0 1  .  In other words, we see that once we quotient by I, we have    t11 t12 t21 t22  (n)  −1   =  t11 t12 t21 t22  (n)  .  4. Finally we have H(Mq (2)) = B/I = n∈N0 Bn /I. The question is of course to know whether this Hopf envelope corresponds to another bialgebra we know. The answer is given in the following remark and proposition. Remark 3.2.6. i) In what follows we are going to look at the quantum determinant in different shifts. More precisely we write det(0) q = (0) (0) −1 (0) (0) t11 t22 − q t12 t21 and then (0) (0)  (0) (0)  (0) −1 det(1) q = S(detq ) = S(t11 t22 − q t12 t21 ) (1) (1)  (1) (1)  = t22 t11 − q −1 t21 t12 ,  (3.6)  since B1 = Mq (2)op,cop and the antipode is an anti-homomorphism. ii) Since det(0) q is group-like when we apply the relations of the ideal µ ◦ (S ⊗ idB ) ◦ ∆ = η ◦ ε and µ ◦ (idB ⊗ S) ◦ ∆ = η ◦ ε as defined above, (1) (0) (1) we find that det(0) = 1 = det(1) is q detq q detq , in other words detq (0) (2) (1) −1 the inverse of detq . We similarly have that detq = (detq ) . Since (2) (1) both det(0) q and detq are inverses of detq and inverses are unique in a (2) monoid, we infer that det(0) q = detq . By recurrence we then get that (2) (2n) det(0) q = detq = · · · = detq  99  and similarly (2n+1) (3) . det(1) q = detq = · · · = detq  iii) Using the relations defining the ideal I on the generating elements of Mq (2), we find (n+1)  t11 t12 t21 t22 t11 t12 t21 t22  =  ·  t11 t12 t21 t22  ·  t11 t12 t21 t22  (n)  (n)  1 0 0 1  =  =  (n+1)  .  Using the formula for the inverse matrix and taking n = 0 in the above, we conclude that (1)  (0)  (1)  t11 = det(1) q t22 , (1)  (0)  t12 = det(1) q (−qt12 ), (0)  (1)  (0)  (1) −1 t21 = det(1) q (−q t21 ), t22 = detq t11 .  By recurrence we then find that, for any n ∈ N0 , (2n)  (0)  (2n)  (2n)  (0)  t12 = q 2n t12 ,  t11 = t22 , (0)  t21 = q −2n t21 ,  (2n)  (0)  t22 = t11 ,  and (2n+1)  t11  (1)  (2n+1)  = t11  t12 (0)  (0)  = det(1) q t22 , (2n+1)  t21  (1)  = q 2n t12  = −q 2n+1 det(1) q t12 ,  (1)  (2n+1)  = q −2n t21  t22 (0)  = −q −(2n+1) det(1) q t21 ,  (1)  = t22  (0)  = det(1) q t11 .  Proposition 3.2.7. The Hopf envelope of Mq (2) is isomorphic to GLq (2) ∼ = Mq (2)[t]/(detq t − 1). 100  Remark 3.2.8. Note that one could prove that GLq (2) is isomorphic to the Hopf envelope H(Mq (2)) by using abstract arguments, namely by verifying that GLq (2) satisfies the proper universal property. Nevertheless, as our goal is here to illustrate the construction of the Hopf envelope, we are going to use explicitly construct an isomorphism between GLq (2) and H(Mq (2)). Proof of Prop. 3.2.7. In order to show that there is an isomorphism, we define two homomorphisms and show they are inverse of each other; consider the homomorphisms defined by ϕ : GLq (2) −→ H(Mq (2)) (0)  tij  −→  tij  t  −→  det(1) q ,  and, for n ∈ N0 , ψ : H(Mq (2)) −→ GLq (2) (2n+1)  −→ t22 t  (2n+1)  −→ − q 2n+1 t12 t  (2n+1)  −→ − q −(2n+1) t21 t  (2n+1)  −→ t11 t.  (2n)  −→  (2n)  −→ q 2n t12  t12  (2n)  −→ q −2n t21  t21  (2n)  −→  t22  t11 t12 t21 t22  t11  t11  t22  We now prove that ϕ and ψ are well-defined bialgebra homomorphisms and inverse of each other. i) ϕ is well-defined and linear. Indeed, (1) ϕ(detq t − 1) = ϕ(detq )ϕ(t) − ϕ(1) = det(0) q detq − 1 = 0.  Moreover, ϕ is linear by construction. 101  ii) ϕ is an algebra homomorphism. This follows directly from the fact that ϕ is defined on the generators of GLq (2). iii) ϕ is a coalgebra homomorphism. We have (ϕ ⊗ ϕ) ◦ ∆(tij ) = (ϕ ⊗ ϕ)(ti1 ⊗ t1j + ti2 ⊗ t2j ) (0)  (0)  (0)  (0)  = ti1 ⊗ t1j + ti2 ⊗ t2j (0)  = ∆(tij ) = ∆ ◦ ϕ(tij ), and (ϕ ⊗ ϕ) ◦ ∆(t) = (ϕ ⊗ ϕ)(t ⊗ t) (1) = det(1) q ⊗ detq  = ∆(det(1) q ) = ∆ ◦ ϕ(t), where we have used that det(1) q is group-like. Concerning the counit we have (0) ε(ϕ(tij )) = ε(tij ) = ε(tij ) by construction of the counit of the coproduct. We moreover have ε(ϕ(t)) = ε(det(1) q ) = 1 = ε(t) since both t and det(1) q are group-like. iv) ψ is linear and well-defined. The well-definition of ψ follows directly from Remark 3.2.6 whereas its linearity is by construction. 102  v) ψ is a bialgebra homomorphism. The morphism ψ is an algebra homomorphism by construction. One (n) moreover checks by direct computation that (ψ ⊗ ψ) ◦ ∆(tij ) = ∆ ◦ (n) (n) (ψ(tij )) and ε(tij ) = ε(ψ(tij )). vi) ψ ◦ ϕ = idGLq (2) . (0) We have ψ(ϕ(t11 )) = ψ(t11 ) = t11 and similarly ψ ◦ ϕ = idGLq (2) for t12 , t21 , t22 . Concerning t, we get ψ(ϕ(t)) = ψ(det(1) q ) 3.6  (1) (1)  (1) (1)  = ψ(t22 t11 − q −1 t21 t12 )  = (t11 t)(t22 t) − q −1 (t21 t)(t12 t) ∗  = (t11 t22 − q −1 t12 t21 )t2 = t, where ∗ uses that t is central and that t12 t21 = t21 t12 . vii) ϕ ◦ ψ = idMq (2)[G−1 ] . Using Remark 3.2.6 we have, for any n, (2n)  (0)  (2n)  ϕ(ψ(t11 )) = ϕ(t11 ) = t11 = t11 , (2n)  (0)  (2n)  ϕ(ψ(t12 )) = ϕ(q 2n t12 ) = q 2n t12 = t12 , (0)  (2n)  (2n)  ϕ(ψ(t21 )) = ϕ(q −2n t21 ) = q −2n t21 = t21 , (2n)  (0)  (2n)  ϕ(ψ(t22 )) = ϕ(t22 ) = t22 = t22 , (2n+1)  )) = ϕ(t22 t) = t22 det(1) q = t11  (2n+1)  )) = ϕ(−q 2n+1 t12 t) = −q 2n+1 t12 det(1) q = t12  (2n+1)  )) = ϕ(−q −(2n+1) t21 t) = −q −(2n+1) t21 det(1) q = t21  ϕ(ψ(t11 ϕ(ψ(t12 ϕ(ψ(t21  (0)  (2n+1)  ,  (0)  (2n+1) ϕ(ψ(t22 ))  (2n+1)  (0)  = ϕ(t11 t) =  (0) t11 det(1) q  =  (2n+1) t22 ;  hence we have just proved that ϕ and ψ are isomorphisms. 103  ,  (2n+1)  ,  3.2.3  Example : Hopf Envelope of a Monoid Algebra  In this section we are going to look at the Hopf envelope of a monoid algebra k[M ]. Remark 3.2.9. We know by Proposition 2.2.28 that for (M, ·, 1) a finite monoid, its monoid algebra k[M ] has a bialgebra structure. Denoting by e1 , ea , eb the basis vectors of k[M ] associated with 1, a, b ∈ M , this bialgebra structure is given by µ(ea ⊗ eb ) = eab ,  ∆(ea ) = ea ⊗ ea ,  η(1) = e1 ,  ε(ea ) = 1.  Knowing that k[M ] has a bialgebra structure, we can now take its Hopf envelope. Construction 3.2.10. The Hopf envelope H(k[M ]) is built as follows. Define, for any n ∈ N0 , k[M ]2n = k[M ] and k[M ]2n+1 = k[M ]op,cop . Then H(k[M ]) = n 0 k[M ]n /I where I is given on generators by (n)  (n)  (n) (n+1) I = (e(n+1) · e(n) − e1 ) a a − e1 , ea · ea  (3.7)  with a ∈ M . Using these relations and the fact that the inverse is unique, we find that −1 −1 e(n+2) = (e(n+1) )−1 = ((e(n) = e(n) a a a ) ) a . We similarly find that (2) (2n) e(0) a = ea = · · · = ea  (3) (2n+1) and e(1) . a = ea = · · · = ea  Proposition 3.2.11. Let M be the group defined by generators and relations as follows. The set of generators is {a | a ∈ M } ∪ {¯ a | a ∈ M } and the 104  relations {a¯ a=1=a ¯a | a ∈ M }. Then the Hopf algebra k[M ] is isomorphic to H(k[M ]). (0)  (1)  Proof. Define ϕ : k[M ] → H(k[M ]) by ϕ(ea ) = ea and ϕ(ea¯ ) = ea with a ∈ M ; we are going to show that it is an isomorphism. i) ϕ is well-defined. Indeed, if a ∈ M turns out to be invertible with ab = 1 = ba then (0)  −1 ϕ(eb ) = eb = (e(0) = e(1) ¯ ). a ) a = ϕ(ea  ii) ϕ is linear by construction. iii) ϕ is an algebra homomorphism. We have (0)  (0)  ϕ(ea eb ) = ϕ(eab ) = eab = e(0) a eb = ϕ(ea )ϕ(eb ). Similarly ϕ(ea e¯b ) = ϕ(ea )ϕ(e¯b ) and ϕ(ea¯ e¯b ) = ϕ(ea¯ )ϕ(e¯b ) . Moreover, (0) ϕ(e1 ) = e1 . iv) ϕ is a coalgebra homomorphism. We have (0) (0) (ϕ ⊗ ϕ) ◦ ∆(ea ) = ϕ(ea ) ⊗ ϕ(ea ) = e(0) a ⊗ ea = ∆(ea ) = ∆(ϕ(ea )) (0)  for any a ∈ M . Moreover ε(ϕ(ea )) = ε(ea ) = ε(ea ) by definition of the comultiplication of the coproduct. We have just proved that ϕ is a bialgebra homomorphism. v) ϕ is bijective. First note that ϕ is clearly surjective. Now suppose that ϕ(ea ) = ϕ(eb ). (0) (0) Then ea = eb and thus ea = eb . Similarly ϕ(ea¯ ) = ϕ(e¯b ) implies ea¯ = e¯b and ϕ(ea ) = ϕ(e¯b ) implies ea = e¯b . 105  Remark 3.2.12. In the previous proposition we have proved that k[M ] ∼ = H(k[M ]). In other words, for a monoid M , its “groupification” M , its monoid algebra k[M ] and H(k[M ]) the Hopf envelope of k[M ], the following diagram commutes H(−) / H(k[M ]) ∼ k[M ] = k[M ] O  O  k[−]  M  k[−]  / (−)  M,  where k[−] is the “monoid algebra” functor taking a monoid to its monoid algebra, (−) is the “groupification” functor and H(−) is the Hopf algebra functor. Remark 3.2.13. We have seen in Section 2.2.4 that k[M ][G−1 ] ∼ = k[M ] for M commutative. Thus, in that case, we get that k[M ][G−1 ] ∼ = = k[M ] ∼ H(k[M ]); in other words, the localization and Hopf envelope are isomorphic as Hopf algebras.  3.3  Weak Hopf Envelope and Relationship with the Localization  In this section we define the notion weak Hopf envelope using the universal property of the Hopf envelope. We also discuss the relationship between the weak Hopf envelope and the localization of a weak bialgebra. Definition 3.3.1. Let H be a WBA, then its weak Hopf envelope is defined as a WHA W (H) together with a WBA homomorphism i : H → W (H) having the following universal property : for any WHA H and WBA homomorphism f : H → H , there exists a unique WHA homomorphism f¯ making the 106  following diagram commute H  /  i  W (H) "  f  f    H.  Remark 3.3.2. The weak Hopf envelope, when it exists, is unique up to a unique isomorphism. Let us now look at the relationship between the localization of a WBA H relative to a monoid of group-likes G and the weak Hopf envelope W (H) of H. Remark 3.3.3. We know by Lemma 1.1.19 that in a WHA all the grouplikes are invertible. Therefore the WBA homomorphism i : H → W (H) factors through the localization H[G−1 ], i.e. we have the following commutative diagram ϕ / H[G−1 ] H i  #    ¯i  W (H), where ¯i ag = i(a)i(g)−1 . As a consequence of the universal property of the weak Hopf envelope, we see that if the localization H[G−1 ] turns out to be a WHA, then it is isomorphic to the weak Hopf envelope W (H), in other words, H[G−1 ] ∼ = W (H). Example 3.3.4. We have seen in Sections 2.2.4 and 3.2.3 that for a commutative monoid M , we have an isomorphism k[M ][G−1 ] ∼ = k[M ] ∼ = H(k[M ]) between the localization and the (weak) Hopf envelope. By the previous remark, we can interpret this as follows. Since M is a group, its group algebra k[M ] has a Hopf algebra structure. Therefore, the localization and the (weak) Hopf envelope are isomorphic.  107  Remark 3.3.3 provides a new description of the localization H[G−1 ] when it is a WHA. The question that naturally arises then is to know whether the localization of H relative to the monoid of all group-like elements G is always isomorphic to the weak Hopf envelope of H. As we shall see in the next example, the answer is in general no. Example 3.3.5 ([Rad80, Example 2]). Consider the two-dimensional vector space V . We can construct on V the coalgebra structure dual to C, viewed as a 2-dimensional algebra over R, in the following way. Let {1, i} be the basis of V . Define the comultiplication by ∆(1) = 1⊗1−i⊗i and ∆(i) = 1⊗i+i⊗1 and the counit by ε(1) = 1 and ε(i) = 0; then V has a coalgebra structure. The tensor algebra H = T (V ) = R ⊕ V ⊕ (V ⊗ V ) ⊕ · · · has then a bialgebra structure and does not have an antipode. The element 1H is the only grouplike and is obviously invertible. Since in this case the monoid of all group-like elements of H is just G = {1H }, which clearly satisfies conditions S1 and S2 of Proposition 2.1.5, we can construct the localization H[G−1 ]. It is immediate that H[G−1 ] ∼ = H and thus the localization is a (weak) bialgebra. This implies that the localization H[G−1 ] does not agree with the (weak) Hopf envelope W (H).  108  Conclusion In this section we discuss the results obtained in this dissertation as well as the new questions they open. The central goal of this thesis was to define and construct the localization of a weak bialgebra relative to a suitable set of group-like elements as well as to study its properties. This question was addressed in Chapter 2; indeed, Theorem 2.3.5 proves the existence and gives the structure of the localization in detail. Together with Proposition 2.1.5 it provides the technical conditions on the group-like elements for the localization to exist. Through the various examples presented, we now better understand the properties of some already known bialgebras and weak bialgebras. With this new construction, several new questions arise. On a technical level, one potential improvement of our theorem would be to “weaken”, or even completely drop, the condition Ig (G) ⊂ G for every element g in the monoid G. This could probably be achieved by studying the properties of the morphism I in detail. Similarly to the bialgebra case, it would be very interesting to know when this morphism is the identity. As we have seen in Section 3.3, the localization of a weak bialgebra relative to the monoid of all group-like elements coincides at times with the weak Hopf envelope. The natural question that comes up is to know pre-  109  cisely when this holds, in other words, to establish a sufficient condition on the weak bialgebra for the localization and the weak Hopf envelope to be isomorphic as weak Hopf algebras. This question is so far open, and being able to answer it would greatly improve our understanding of both the localization and the weak Hopf envelope. Concerning the weak Hopf envelope, we have presently no construction that holds in general. It would thus be very interesting to develop such a construction in the weak case. This would moreover enable us to study its properties in a precise way. In a broader research program this could be applied to the recently developed combinatorial characterization of fusion categories of [Pfe11] in the following manner. Consider a finite graph G and the WBA structure on the path algebra H[G] = k(G×G) of the graph G×G. Then quotient this weak bialgebra by relations of the form RT T − T T R obtained by a solution R to a weak generalization of the quantum Yang-Baxter equation (as in [Pfe11]). Next, taking a finite-dimensional quotient of the weak Hopf envelope, we could systematically construct numerous new fusion categories. This method would moreover provide a completely new approach to the difficult question of the classification of these categories.  110  Bibliography [Bea99]  John A. Beachy. 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