Colloquium talk at IMSc by Ben Elias on Friday 01 Feb, 2014
1400--1500 hrs, room 326
Title: The new homological algebra:
p-complexes and categorification at roots of unity

Abstract:
Homological algebra has been at the foundation of the modern study
of topology, representation theory, and geometry. Complexes,
homotopies, derived categories, and dg-algebras are powerful tools
and are, for many, a way of life. However, in a 2005 paper, M.Khovanov
observed that homological algebra as we know it is but a
theory attached to the finite (super) Hopf algebra k[x]/x^2, and
that this whole framework can be generalized to any finite dimensional
Hopf algebra! He called the resulting theory "Hopfological algebra,"
and it was developed further by Y.Qi.

One special case holds particular interest: the (non-super) Hopf
algebra k[x]/x^p over a field k of characteristic p. For this Hopf
algebra, one should consider p-complexes, which are like ordinary
complexes except that d^p=0 instead of d^2=0, and the tensor product
rules have no signs. It turns out that many interesting algebras
appearing in geometric representation theory can be equipped with
p-differentials, so that they become p-dg-algebras. Interestingly,
the homological shift acts on the Grothendieck group of a p-dg-algebra
by multiplication by a p-th root of unity (instead of multiplication
by -1). In this fashion, we can transform many of the recent
categorifications of quantum groups and their representations into
categorifications at roots of unity.