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Primitive idempotent elements of a total matric algebra Poole, Albert Roberts 1931

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P R I M I T I V E  I D E M P O T E N T o f  T O T A L  M A T R I C  E L E M E N T S  a  -  A L G E B R A .  b y  ALBERT ROBERTS POOLE  I U . B . C . LIBRARY j g CAT  |  P R I M I T I V E  I D E M P O T E N T of  T O T A L  W)  L.Spfi ..(iijJs^._PbP<i  E L E M E N T S  a  M A T R I C  A L G E B R A .  b y  ALBERT ROBERTS POOLE  A T h e s i s submitted f o r the Degree of MASTER OP ARTS i n the Department of Mathematics  THE UNIVERSITY OP BRITISH COLUMBIA February, 1931.  n  Table of Contents  page Introduction •  •  •  •  •  •  •  •  •  •  •  •  •  •  I  Part I Primitive idempotent elements of a total matric algebra  1  Part II Transformations which leave the multiplication table of a total matric algebra invariant Bibliography  .  •  .  *  •  .  9 S3  I  Introduction P a r t I of t h i s t h e s i s i s concerned w i t h determining  the problem of  n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s , t h a t an  ment of a t o t a l m a t r i c  algebra  over a f i e l d  r  ele-  shall  be a p r i m i t i v e idempotent. A supplementary s e t of p r i m i t i v e idempotent elements i s d e f i n e d , and necessary and c o n d i t i o n s t h a t TV  sufficient  p r i m i t i v e idempotent elements form such  a s e t are d e r i v e d . P a r t II of the t h e s i s c o n s i d e r s mining a l l t r a n s f o r m a t i o n s t a b l e of the a l g e b r a  the problem of d e t e r -  which leave  the m u l t i p l i c a t i o n  i n v a r i a n t . For t h i s purpose  supple-  mentary sets of p r i m i t i v e idempotents are employed. The t h e s i s was w r i t t e n under the s u p e r v i s i o n of P r o f e s s o r P. S. Howlan of the U n i v e r s i t y of B r i t i s h Since  the r e s u l t s were completely  Columbia.  determined by him p r i o r to  a s s i g n i n g the t h e s i s s u b j e c t , he r e s e r v e s  the r i g h t of pub-  l i c a t i o n under h i s name. The m a t e r i a l of the t h e s i s however, developed without r e f e r e n c e  to h i s work.  was,  1.  PART I  Primitive Idempotent Element a of a Total Matrio Algebra. Let Al be a total matrio algebra of order field  F .  Let a set of basal units of f\  then any element of  over a  be  i s of the form  where 'K*,*^, C^>#^ > - ~ " "™) l  i  s  i  n  We f i r s t determine necessary and sufficient conditions that an element of - 21  , -e^- .  shall be an idempotent* i s an idempotent, then  If  ^  , so that  whence, since the -£-_*.y^ are linearly independent, we have f 1)  ^  J - ' , - - - " ' ) .  ^  Conversely, i f ( 1 ) holds, then  i s an idempotent. We  therefore have Theorem 1 .  An element  =£  c^^-e^.^  0  f  a  total matric  algebra i s an idempotent i f , and only i f , | : ^ ^ = ^ c ^ i - - " » » ) . We next determine necessary and sufficient conditions that an element of ^  =  i^^V^'^  shall be a primitive idempotent. *  s  a  If  Pri ^!" ® idempotent and i f 10  7  i s any element of Af which makes the element-^^ctt of the a l -  gebra-cX'AI^ an idempotent, i t follows  th&t ^ c u - ^  o on tain a no idempotent but  •**-y  sinoe  t  •  We determine an element yc •=  , of A1  -^u:^,  which w i l l make-^cU' an idempotent.  i s to be an idempotent i t follows from  I^U^UJU  (1)  that we  must have  Since ^  i s an idempotent we employ (1), and we see i f a ^ o o ^  i s to be an idempotent i t i s necessary that /  ,  -  . oi  /3  oi  , 2L  otf'.  /3  ^  -  This condition i s satisfied i f we take •2  ^  that i s , i f we take That o<^^=  /^L^ =  ^  or /3L~*e  —/ = ^^/-^-v^.^ -  has an inverse when / f ^ ^ ^> ^  the f orm ,  »^  o  2  n  y  l  w  ^ _^ ^*'^= 4- - -  i  n  "^L^ -  f  e  *  ^  e  •  Therefore i f we choose  ^ ^ ^ y = / -  > with at least one  then-/X"?cx6 w i l l be idempotent. As a special case take  i s evident since  >o)  of  i s either zero or  ^ S ^ = >™) not zero*  3. Then  Bat  -<^cc^  , therefore  -  Whence f o l l o w s  ^  i n which  o  condition that-^o Mow l A\  e  t  .  R e l a t i o n (a) i s t h e r e f o r e a n e c e s s a r y  i s a p r i m i t i v e idempotent. -  and l e t r e l a t i o n  °*--Lg ^  he an idempotent element o f  fa) h o l d .  Let  any other element o f A l .  fi-Cg^j,  be  Then  and on employing ( S ) , we have  ^  where -A! i s a number o f the f i e l d . order  =jE/  Therefore - u ^ A l ^  i s of  , and so c o n t a i n s no idempotent element except c ^ o .  Accordingly-^  i s a p r i m i t i v e idempotent, and we see t h a t (S)  i s a s u f f i c i e n t c o n d i t i o n t h a t an idempotent element s h a l l be primitive. have  From t h i s r e s u l t w i t h referenoe  t o Theorem 1, we  Theorem 2. An element  of A l i s a primiti-  *Jp-^g  ve idempotent element i f , and only i f ,  Prom (1) and (2) we have  Since not a l l the ^  /, - -- -wj are zero, we have  Relations (1) and (2) can therefore be replaced by relations (2) and (3), since (3) follows from (1), and (1) from (3) on employing (2). Corollary 1«  We therefore have^ An element  -  i s a primi-  tive idempotent i f , and only i f ,  and  J~r ^  =  7  The above corollary leads to a simple method of writing down a primitive idempotent element of a total matrio algebra of x  order -?v •  Construct an array withTt' rows and -n^o columns  as follows: Write in n u m b e r s ( * *  ,-^>) along the principal  diagonal, so that their sum i s unity. column a r b i t r a r i l y .  Then f i l l up the f i r s t  Choose the numbersoi. (-:-- ^- - -  of  the second oolumn. so that the r e l a t i o n - ^ ° ^ x i ° ^ x 6 * - - i ' ' > ^ ) i s satisfied, and in general choose the elementsa^ in the column, so that the relation ot--,°i.° ^^^i/^'  -H) '• ~~ ~  i s satisfied. To show that an element i s a primitive idempotent when i t s coordinates are chosen in the above manner, we must prove that relations (&) and (3) hold.  Relation (3) evidently holds  since the sum of the diagonal numbers i s unity.  That (2)  holds i s shown as follows: i s defined by the relation ?  which i s also true for  3*'  Therefore °Q fay  ~) satis-  fies the relation (a)  = /  ^•vw satisfies (b)  •it/  satisfies (o)  and cO^., satisfies (d) It readily follows that  To illustrate we write down a primitive idempotent element of a total matric algebra of order 16.  6. /  -/  — z  a.  _  3 5L  3  3  ** 6  The element i s then  Definition.  A set of primitive idempotent elements w i l l be  called supplementary in case their sum equals the modulus, and i f further the product of each pair in either order i s zero. It can be proved" that there are -ru> elements i n a sup1-  plementary set. We derive necessary and sufficient conditions that a set of TJ  primitive idempotent elements of A\ shall form a supple-  mentary set. Let  ^  *  1  <^  lowlan. B u l l . Amer. Math. Soc.  ™)  A p r i l , 1930.  p. S68  7.  be a supplementary set of primitive idempotent elements. Then  That i s  ^  —  ^  Therefore  and we have  Again  that i s Z-/  *e-  = z ^ - ^  and we have ^  / j r i  (5) and  Conversely, by retracing our steps,  it u<Cg  ^ 'j -  -—»^)  are-TIP  primitive idempotent elements and i f (4), (5), and (6) hold, then ^ c ^ ,  /,  j-™) i s a supplementary set of primitive  idempotents.  We therefore have  8*  Theorem 3.  The w  primitive idempotent elements C''  ~;  constitute a supplementary set  of primitive idempotents, i f and only i f ,  and  PART I I Transformations Whioh Leave the M u l t i p l i c a t i o n Table o f a T o t a l M a t r i o A l g e b r a At  Invariant.  seek t r a n s f o r m a t i o n s whioh w i l l c a r r y the - e ^ ^ u ^•-/,- -yy)  We  i n t o new b a s a l u n i t s  ^**>^ = 4-->*^for whioh  and  L  e  -  t  ^  = '  ^  be a supplementary s e t o f p r i m i t i v e idempotent elements o f the t o t a l m a t r i c a l g e b r a Let  denote a c e r t a i n f i x e d non-zero c o o r d i n a t e  o f the p r i m i t i v e idempotent element  (w - /,  >  •  That such a non-zero c o o r d i n a t e e x i s t s f o l l o w s from ( 3 ) . Let  ^  = H  ^s^-c^  be  any  element o f A |  .  0)  - Z  But from (a) we have  or  <*\  - J ^ ^ ^ L  since <  ^  ^  10.  Employing this, we have  Observing that L^<^^  <  ~  ~ ~ ~> ""y  i s independent of ;?r and noting that so far no use has been made of the faot that the primitive idempotent elements £ - • - . > ^) Theorem 4.  form a supplementary set, we have and^^jare two primitive idempotent ele-  ments of a total matric algebra A\ then the algebra-U^ CA i s of order  1  Since /A  • i s simple* and s i n o e . ^ ^ ^ - ~ - ^ ) i s a supple-  mentary set of primitive idempotent elements we have  Dickson: Algebras and Their Arithmetics, p. 80. Dickson:  Algebras and Their Arithmetics, p. 73.  2  11. where A/^.denotes that  -vo) . i s of order  /  We have proved  , and we oan therefore  write any element of A!^in the form/^-*^/ where & ranges over F  and  i s given by  Prom (7) we have  where X? and  range over '  where /b  &  t  %  •  Also from (8) we have  = ^  and ^  ^  ^  F  range over  We shall determine a value for y  *- '  ^  . following a suitable  choice for /S and & such that  The elements  w i l l then have tne multiplication table of a total matric algebra.  Combining (9) and (10) we require  * - < * - £ =  <  «  &  /3  M ^fsL  ^^*^-V  «  S ; ; Y  or, on simplifying, by use of (2) and (3),  •ftefit  >  J J  12.  This condition i s satisfied i f we take  and  f^t%  f  We therefore have /  for which  ^ ^ = ^ ^ ^ ^ ^ - - / / v - J 0  Expressing -^-^^  ^  "  ^  ^  ^ . ^ - h ) *  V  ^ = /; - - - ,-^)  ^  in terms  oi~e^^(^^=  - ;-»^)  we have the transformation  This transformation w i l l hereafter he referred to as Type I. To show that the•  £W  /;—  are a set of basal  unitB we mast prove them linearly independent.  The determin-  ant of a transformation of Type I is of order -yo denoted by  —^^r,  ( y y ^ - i ) ^  \  -fa -P  ~fzcf  and column  and can be  in which  i s the element belonging to row  '•- /) ^  ~*~-^  //  •  W e  shall use the  symbols ^ - ^ a n d ( ^ ^ ) to designate the above row and oolumn respectively.  13. To investigate the value of the above determinant, we use the following Lemma.  » power of the product of two "v^ » order de-  The  terminants /) and /3 may be expressed as a determinant C> of order -r^  which may be obtained by replacing the element  of the  row and ^t- »  »  column of rt  by the elements in  their natural order obtained on multiplying each element of by the element of the •i^f' row and  /3  ^  column of  The method of proof being perfectly general, we take •n.= £ whenever i t i s necessary to i l l u s t r a t e . -  and  /3  & j  =•  -  Let  'VOX,  = '>  Then Also the -TV y power of any  -» order determinant may be  expressed as a determinant of order -yx> in the form i l l u s trated below: CU  a.  a,  ex.  ft 3/  a,  au  31 33  a.  «^ „ (St-  a.  a,  0  o  o  o  o  O  o  o O  o  o  o  o  o  o  o  o  o  o  o  o  ex. '3  3/  o s>  o o  31  o o  *  o  O O 3  o <%,  ^  14. /  Denote the determinant above on the right, by n  3  , S3  w i l l equal a similar determinant in which - 6 ^ 6 ^ ~~ C- ~ replaces CL-  ; denote this determinant by &  fr^-  In the determinant/?  interchange t h e ^ ^ ) row with  rowfe-^*i,- "*\) and treat the determinant »  and  R-^r')n  /3 = (rO  , where  ft  and &  and /S  and  (fta)  =  respectively.  /?  /?  =  •  i s the number of  S  interchanges of rows involved in changing ft and /?  &  Then  =  For the i l l u s t r a t i o n ,  a. ou  as  c o o  o  a. as a.  o 0  0  •1 ii  °i, 0 0  ° 0  0  0  0  0  o  0  o ex.  o °  a- a. 11  O  0  0  o  11  -6  /I  O  •£  o  /3  a.  11  a  O  0  O  o  o  0  0  O  o O  O O  -c^,-*,,  0  e> o O  o  0 - * , i ^ , 3  a. „ <x IZ '3  °  ° 9  0  i3  o  0  o  o  o  o  o  O  XI It 1 13  O  O  O  O  O  Si  a  a. 33  0  o  o  o  o  o  ti  <7  O  C&*J  in the same manner.  Denote the determinants thus obtained by/? Then  the  .  O  O  O  -§^3  into  a,  4  ti  - ii  CL4 ii  f /«. /' Ij  a,  i t  a,  -e  3?  ii  3/  CK.  11  a.  S  ti  -g  ia.  ~g  (X.  -4  (X-  —^  f3^t3  /1 «. a.  1  ex.  31  •3/  a.  .  "  °31  a. Z  l  ^  3/  71  t-  'Ti i  J l  A-  4  a,i t-/ t j  m  11  CL. -Q  3 _?Z  31 cc  31  '  -Z 3 CU  -&  73 /'  s^rii  3  33  J  3'  1?  a,  </  -&  3S  CK.  3-2.  3*  31  11  -f> 1  This determinant is of the form £  and therefore the Lemma  is proved. Conversely, by retracing the steps of the proof, we see that any determinant of the form C< can be replaced by ft  /2> ^fSft/sj  where  ft  and fo represent ^ >, order determin-  ant s. Observing that the determinant of a transformation of Type I i s of the form duces to the  of the Lemma, we note that i t re-  » power of the product of the determinant  In the f i r s t of these determinants and us the column;  designates the row  in the second determinant  designates  the row and ~j/ the column • Interchange corresponding rows and columns in the f i r s t of these determinants so that-*-  denotes the row and  the  16, oolumn.  Then consider the product  (I)  The element in the -r" » row and  S » oolumn of the resul-  tant determinant is  Applying (4), this reduces to aero unless V ~ S  cing  t  Pla-  and applying (a) and (3), we have  The resultant determinant has therefore a l l elements in the principal diagonal equal to unity, and a l l other elements equal to zero.  It i s therefore of value unity.  The determinant of a transformation of Type I is the -v^,] power of this determinant and so equals unity.  The - ^^/6~> t  &/  are therefore linearly independent and so are a set of basal units.  Also the relations J  ^  and  hold.  >  -»"  v v  ^  — o  A transformation of Type I i s therefore of the type  sought. It i s evident that there are many different transformations of Type I, due to the possibility of different supple-  17. mentary sets of primitive idempotents. Theorem 5.  We have proved  Any transformation of Type I leaves the multipli-  cation table of a total matric algebra invariant and has i t s determinant equal to unity. We now proceed to determine whether there are transformations other than those of Type I which leave the multiplication table of a total matrio algebra invariant. ~L  L e t  i^l)  be any non singular transformation which leaves the multiplication table of a total matrio algebra invariant. - c ^ ^ ^ ^ c  /_ - -yvJ)  i  s  a  Then  supplementary set of primitive idem-  potent s. Also  whenoe fl3)  =  fa,  ^  <AA  '->" ~  From this, on employing (2), we have  -fata  '  which, by means of (13),  K ~ r ^  yields  J  . (14)  °<-_^,  18. u-;  ~-—^ee)  J  Therefore transformation (12) can he written as  -fat-fat  /i^v. / ^t^v-'  let  /  (16) It i s seen that the-ct^y  /, -v~J are a set of basal units  obtained from the - ^ ' - ^ 6-*^ ~ ' 7 ~ transformation (16) which i s of Type I, The 4 ""^ * obtained from the C*^ — - ^ b y the transformation a  Whenm^  =  we have ~ ^ ^ c  w  =  r  e  -^^^^^A~-^»^j£ransformation  h e n  (17)  i s therefore a transformation which leaves the multiplication table of a total matric algebra invariant and which also leaves the JUU.^  /yi-~/.  .-  invariant. by C~l  Denote  0^  s  t -—J ~"J•  and (17) i s of the form  (is)  ^Cu -  It i s evident that none of the  C~~j^~(>--'<•*) • '> ~~>^)are zero,  Y^£(^-,  because we hove shown that the ux^j  //  are a set of  19. basal units, and we have chosen the -<£^/ (-™*  /, - - -  as  a set of basalvunits. We have  which gives  Also  which gives  .  i  Combining these two results we have  where  <)C~~~  =  y  ^"""^ ^ 4 " ~ "^J as  a  necessary condition (-^ ^ = o - -  that transformation (18) transforms the into a new set of basal units the  *4—•*-«-J  which  have the multiplication table of a total matrio algebra, and are such that  ^ -  UMJ^JU  ^  ~ 1 ~ ~ ~  It i s evidently a sufficient condition.  •  We shall here-  after refer to transformation (18) with the accompanying relation (19) as Type II. Since from (19) we have  so. we have evidently Theorem 6.  Any transformation of Type II leaves the multi-  plication table of a total matric algebra invariant and has i t s determinant equal to unity. We have also proved Theorem 7.  Any transformation ~f which leaves the multipli-  cation table of a total matric algebra invariant may be written T -  where 7^ ant of T  7Z77  i s of Type II, and 77 i s of Type I .  The determin-  is equal to unity.  We now prove Theorem 8. Any transformation "7^ of Type II may be written where S,  and  are transformations of Type I.  Multiply 7J~ on the l e f t by transformation ^ which w i l l be constructed as follows: Let  = ZLv  ^ >  ^*" "' *  be a supplementary set of primitive idempotents,  J  J  so chosen,  as i s always possible, that the non-zero coordinate  °^A-A,  '~ *> ~"  * ^  e  toattsformation o>,  i s then  21. We have  which on employing (19) "become^^^.j  ggfj  The determinant of this transformation i s the product of the determinant of ^  by that of  and so equals unity.  Also  and ^"leave the multiplication table of a total matric algebra invariant, and therefore so does<5>! ^^i> ~  •  Hence  i s a set of basal units of algebra A l  such that  and  .  Therefore the  ^-***- =^-->*Oare a supplementary set of  primitive idempotents.  From f20),  (19),  "  (2) and (3) -7—  It i s then evident that transformation^ ^Tcan be considered as a transformation  of Type I, formed from the supple-  mentary set of primitive idempotents That i s whence  _  ;  (-™~, ~- 'j ~ - >  ,  as* Combining Theorems 7 and 8 we conclude with Theorem 9. Any transformation ~T~ whioh leaves the m u l t i p l i cation table of a total matric algebra invariant may be written  _/  where  ,  S,  and fi, are transformations of Type I, and  ~7£ i s of Type I I , The transformation T unity.  has i t s determinant equal to  S3.  Bibliography 1.  L . E . Dickson:  U. S. A . 2.  t  Algebras and Their Arithmetics.  Chicago,  July 1923.  F . S. Howlan:  On the Direct Product of a Division A l -  gebra and a Total Matric Algebra. Mathematical Society.  New York,  Bulletin of The American April 1930.  

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