Schedule, Notes/Handouts & Videos

Many noncommutative algebras in representation theory and combinatorics arise naturally as twisted tensor products: they decompose as a vector space into the tensor product of two subalgebras.  Examples of twisted tensor products and their deformations include skew polynomial rings, quantum planes, Weyl algebras, Ore extensions, universal enveloping algebras, quantum Schubert cell algebras, smash and crossed products (for group/Hopf actions), braided tensor products defined by R-matrices, some Sklyanin algebras, some Sridharan enveloping algebras, graded affine Hecke algebras, and symplectic reflection algebras.  Often a Poincare-Birkhoff-Witt property, triangular decomposition, or particularly fruitful application of the Diamond Lemma points to the underlying structure as a twisted tensor product.  We discuss resolutions for twisted tensor products and give Alexander-Whitney and Eilenberg-Zilber maps for converting between resolutions.  These maps allow one to transfer abstract homological information into concrete conditions for determining deformations of algebras.

We will consider representations of the Lie algebra of differential operators of order at most one on the algebra of complex Laurent polynomials.  This Lie algebra contains a natural family of large subalgebras that arise from certain pairs of Laurent polynomials.  We will examine representations induced from these subalgebras, and we will see that many are in fact irreducible representations.  Many of the key ideas follow from some general results concerning tensor products as well as from the fact that these subalgebras have finite codimension.

Braid group actions on vector space tensor powers are used to define a large class of associative algebras, termed Nichols algebras. These Nichols algebras underlie the structure of many Hopf algebras, including the small quantum groups, and are essential in classification theorems for some types of finite dimensional Hopf algebras. The structure of Nichols algebras, e.g. PBW-type bases in some settings, is governed by braiding. This facilitates homological techniques for understanding representations. We will introduce cohomology of finite dimensional Hopf algebras, and state an open finite generation conjecture. We will describe some recent results for Hopf algebras arising from Nichols algebras.

We consider reduced imaginary Verma modules for the untwisted quantum affine algebras $U_q(\hat{\g})$ and define a crystal-like base which we call imaginary crystal base using the Kashiwara algebra $\mathcal K_q$ constructed in earlier work by Ben Cox and two of the authors. We prove the existence of the imaginary crystal base for any object in a suitable category $\mc{O}^q_{red,im}$ containing the reduced imaginary Verma modules for $U_q(\hat{\g})$.

Come join us for a special evening celebrating the remarkable achievements of Georgia Benkart. The banquet will be held at the Hong Kong East Ocean Seafood Restaurant, where we will gather together to show our appreciation for Georgia and all she contributed to the mathematical community.