On the twist-equivalence of certain quadratic algebras associated to finite-irreducible Coxeter groups

• Main Author: Dold, Colin
• Other Authors: Stafford, Toby ; Bazlov, Yuri
• Published: University of Manchester 2016
• Subjects: 512
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.686777

This is the author's PhD thesis. During the course of my studies, I have mastered the basics of the theory of Hopf algebras, and in particular, learned of Drinfel'd's concept of a cocycle twist of a Hopf algebra, and of a module algebra over that Hopf algebra. A module algebra is is an algebra over a field upon which a Hopf algebra acts in a certain way. In particular, I came to focus upon how this concept works in the special case when the Hopf algebra is an algebra over a finite group. In this case, the module algebra is an algebra over the field on which the group acts by homomorphisms. This algebra may be twisted into another module algebra by means of a 2-cocycle on the group. Having learned this, my attention was drawn to the work of Vendramin in [Ven12] in which he examined two module algebras over the symmetric group S_n, called Nichols algebras, defined using what are called rack cocycles. He showed that there is a cocycle twist that transforms one algebra into the other, i.e. that they are twist-equivalent. There are two quadratic algebras associated to the Nichols algebras, called E_n and Lambda_n and first described in [FK99] and [Maj05], which are thought to be isomorphic to the Nichols algebras. It has for some years been conjectured, but not proven, that these two algebras are twist-equivalent. The most important result of this thesis is Theorem 4.6, which proves that E_n and Lambda_n are indeed twist-equivalent. Following this result I sought to see if analogous results could be obtained when considering other finite irreducible Coxeter groups than type A, which is what S_n is. To do this requires understanding of rack cocycles, and of the Schur multiplier of a group, which affects what kind of cocycle twisting is possible. I chose to focus on the case where the Coxeter group is a dihedral group since these groups are often fundamental to determining what happens for Coxeter groups of higher dimension. The last part of this work examines questions on whether the rack cocycles analogous to those that defined E_n and Lambda_n are related to each other by cocycle twisting. The dihedral case, however, turns out to be less straightforward than was the case for Coxeter groups of type A, and it seems that there is scope for continuing research in this direction.