Manchester Geometry Seminar 2014/2015

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.)