[Mathsem] Maths_Colloquium on Friday, Jan 31, 1400 hrs, rm 326

[K.N.Raghavan]

Mathematics Collquium at IMSc
Note: Unusual day, unusual time, and unusual venue
Friday, January 31 2014
14:00 - 15:00
Venue: Room326

Title: The new homological algebra: p-complexes and categorification at roots of unity
By: Ben Elias
From: Massachussets Institute of Technology
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.
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K. N. Raghavan | email: k...@imsc.res.in
Inst. of Math. Sciences (IMSc) | phone: +91-44-2254 3264/3166
C.I.T. Campus, Taramani | (direct/board)
Chennai 600 113 | +91-44-2844 1505 (home)
India | mobile: +91-944-563 1505
| fax: +91-44-2254 1586 (work)
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