UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Localization in weak bialgebras and Hopf envelopes Bennoun, Steve


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, [formula omitted] 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 GL_q(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. 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.

Item Media

Item Citations and Data


Attribution-NonCommercial-ShareAlike 3.0 Unported