A tensor category is a category equipped with a nicely behaved tensor product, for example take Rep(G) where G is a finite group. In this talk I will explain how to do calculations in tensor categories by manipulating certain pictures. I will also explain how various diagram algebras (such as Temperley-Lieb or Brauer algebras) arise as endomorphism algebras in some "universal" tensor categories.
Diagram algebras from the viewpoint of tensor categories
Apr. 18, 2011 2pm (MATH 350)
Lie Theory
Rahbar Virk (UC Davis)
X
The category O of (modules of) a complex semisimple Lie algebra was introduced by J. Bernstein, I. Gelfand and S. Gelfand in 1976. Since then it has found applications and motivated much progress in areas as diverse as knot theory, D-modules, the geometric Langlands program and combinatorics. I will explain how much of the structure of category O can be explained by viewing it as a `reasonably faithful module' for the Weyl group (or rather the associated Hecke algebra). I will situate myself on the algebraic side of this story, make this statement precise and present some of my own contributions to the subject. To keep things as elementary and concrete as possible I will mainly focus on sl2 and sl3. Time permitting, I will say something about the algebraic geometry underlying this story.
A supercharacter theory is an approximation to the usual character theory of a group. Even very coarse approximations still contain useful information about the group and have been exploited in situations when the full character table is unknown. We describe some of the ways that supercharacter theories arise in the context of the general linear group over a finite field.
Supercharacter Theories of GL_n(F_q)
Aug. 30, 2011 2pm (MATH 350)
Lie Theory
Fred Goodman (Iowa)
X
The Birman--Murakami--Wenzl algebra (or BMW algebra for short) plays an important role in knot theoretic invariants and in centralizer algebras for quantum groups. It may be defined either by (rather complicated) generators and relations, or as an algebra of diagrams with a pictorially defined multiplication. Affine and cyclotomic Hecke algebras are related to the Hecke algebras of type A in the same way that the wreath products Z \wr S_n and (Z/rZ) \wr S_n are related to the symmetric group S_n. The affine and cyclotomic BMW algebras discussed in this talk are BMW analogues of affine and cyclotomic Hecke algebras.
Diagram algebras from the viewpoint of tensor categories