Workshop on Lie Groups, Lie Algebras and their Representations

Abstract. In the 1970s, Bernstein, Gelfand and Gelfand proved a very important reciprocity relation in the category $\mathscr O$ of modules for a simple Lie algebra. Their result related projective modules, Verma modules and irreducible modules in the categroy.

In this talk we will be concerned with an analogy of BGG duality for for the current algebra $\mathfrak{sl}_2[t]$ of $\mathfrak{sl}_2$, or equivalently the maximal parabolic subalgebra of the affine algebra. We define the analog of the category $\mathscr O$ in this context, roughly speaking, the category of modules for $\mathfrak{sl}_2[t]$ which are sums of finite dimensional modules of $\mathfrak{sl}_2$. We shall introduce the simple modules and their projective covers, define the appropriate analogs of the Verma modules and finally obtain the required duality.

This is based on joint work with V. Chari and N. Manning.

Abstract. Weyl modules were defined and studied over the last two decades in various cases and from various perspectives. In this talk, I'll extend the definition of local Weyl modules to equivariant map algebras (EMA). By showing that there exists an isomorphism between the category of finite-dimensional modules for the EMA and the analog for the generalized current algebras, one can translate every reasonable question about this category in the EMA setting into the context of generalized current algebras. This applies especially to Weyl modules and irreducibles.

Abstract. A new class of simple perverse sheaves was defined recently by Zheng in categorifying the tensor product of simple modules of a quantized enveloping algebra. In this talk, I will present my recent work on the interaction between certain class of tensor product varieties (in the spirit of Lusztig, Malkin and Nakajima) and this class of simple perverse sheaves. I'll discuss in details the structures of the classes of varieties and perverse sheaves involved. This leads to the equivalence between the localization process using support of complexes and the localization process using singular support of complexes.

Abstract. Suppose a finite group acts on an algebraic variety $X$ and a finite-dimensional Lie algebra $\mathfrak g$. Then the space of equivariant algebraic maps from $X$ to $\mathfrak g$ is a Lie algebra under pointwise multiplication. Examples of such equivariant map algebras include current algebras, twisted and untwisted loop algebras and their multi-variable versions, and the (generalized) Onsager algebra.

In this talk I will present a classification of all finite-dimensional irreducible representations of equivariant map algebras (joint work with Alistair Savage and Prasad Senesi) and describe their extensions (joint work with Alistair Savage). The latter result allows us to determine the block decomposition of the category of all finite-dimensional representations.

Abstract. We will discuss the theory developed with Joe Chuang of 2-representation theory. We will explain how representations of Kac-Moody algebras are replaced by actions on categories. We will explain the theory of highest-weight 2-representations.

Abstract. I will describe monodromy representations of affine braid groups arising from a flat connection with values in the Yangian of a simple Lie algebra $\mathfrak g$. These representations are related to those arising from the quantum Weyl group operators of the quantum loop algebra of $\mathfrak g$. Matching these two classes of representations involves in particular the construction of a functor relating finite-dimensional modules of those two quantum groups.

This is based on joint work with Sachin Gautam.

Abstract. It's a long established principle that an interesting way to think about numbers as the sizes of sets or dimensions of vector spaces, or better yet, the Euler characteristic of complexes. You can't have a map between numbers, but you can have one between sets or vector spaces. For example, Euler characteristic of topological spaces is not functorial, but homology is.

One can try to extend this idea to a bigger stage, by, say, taking a vector space, and trying to make a category by defining morphisms between its vectors. This approach (interpreted suitably) has been a remarkable success with the representation theory of semi-simple Lie algebras (and their associated quantum groups). I'll give an introduction to this area, with a view toward applications in topology; in particular to replacing polynomial invariants of knots that come from representation theory with vector space valued invariants that reduce to knot polynomials under Euler characteristic.