# Manchester Geometry Seminar 2014/2015

Thursday 13 November 2014. The Frank Adams Room (Room 1.212), the Alan Turing Building. 4.15pm

## Bialgebras from Algebra Factorisations Yuri Bazlov (University of Manchester)

yuri.bazlov@manchester.ac.uk

Bialgebras and, more specifically, Hopf algebras are often viewed as vastly generalised versions of group rings and Lie algebras, and provide the necessary formalism for supergroups, quantisation and so on. I will talk about a new way to produce bialgebras. An algebra C which factorises into its subalgebras $A$ and $B$, $C = AB$, gives rise to a certain bialgebra $H$ which gauges the degree to which $A$ and $B$ fail to commute within $C$. Examples of such algebra factorisations include functions on a direct product of two (super)manifolds, semidirect products of groups, and more intricate constructions such as matched pairs of groups, although the original motivation for this work was a class of Hecke/Cherednik algebras which are of current interest in representation theory. Essentially, the bialgebra $H$ is constructed using elementary (not to mean easy) linear algebra. It is interesting to point out that every finite-dimensional bialgebra arises in this way. I hope to survey this new construction and to show how it leads to a new proof of a particular case of the Poincare-Birkhoff-Witt theorem. (Joint work with Arkady Berenstein.)