Lie Theory Workshop on Quantum Groups

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Abstract. The talk will focus on connections between quantized coordinate rings of algebraic groups and varieties and their corresponding classical coordinate rings, in particular, the connection via the Poisson structure on the classical coordinate rings arising from the construction of the quantized coordinate rings. The Orbit Method in Lie theory, which was first developed to parametrize irreducible unitary representations of a nilpotent Lie group $G$ by the coadjoint orbits in the dual of the Lie algebra $\mathfrak{g}$ of $G$, evolved away from actual orbits via the theorem of Kirillov, Kostant, and Souriau, which established that the coadjoint orbits in $\mathfrak{g}^*$ coincide with the symplectic leaves for a standard Poisson structure.
As a general principle, the Orbit Method suggests that the primitive ideals of a noncommutative algebra $A$ should be parametrized by the symplectic leaves in some Poisson variety associated with $A$. For the case when $A$ is a generic quantized coordinate ring of a variety $V$, it has been conjectured that the space of primitive ideals of $A$ is homeomorphic to the space of symplectic leaves in $V$, both spaces being equipped with appropriate Zariski topologies. The extent to which this conjecture, with some modifications, has been established will be discussed.

Abstract. In this talk, based on a joint work with A. Berenstein, we define a topological Hall algebra by dropping the exactness. The resulting algebra is a deformation of the completion of the usual Hall algebra with respect to the grading by the Grothendieck group, and its associativity leads to rather non-trivial $q$-binomial identities. To establish the existence of an integral isomorphism of the topological Hall algebra onto the completion of the Hall algebra, one needs to introduce exponentials of categories and study their factorizations. This leads to an analogue of triangular decompositions for categories and multiplicative identities for the exponentials.

Abstract. There are two approaches to projective representation theory of symmetric and alternating groups, which are powerful enough to work for modular representations. One is based on Sergeev duality, which connects representation theory of the supergroup $Q(n)$ and projective representation theory of the symmetric group via appropriate Schur algebras and Schur functors. The second approach follows the work of Grojnowski for classical affine and cyclotomic Hecke algebras and connects projective representation theory of symmetric groups to the crystal graph of the basic module of the twisted affine Kac-Moody algebra of type $A_{p-1}^{(2)}$. We explain how to connect the two approaches mentioned above and to obtain new branching results for projective representations of symmetric groups. This is achieved by developing the theory of modular lowering operators for the supergroup $Q(n)$ which is parallel to (although much more intricate than) the similar theory for $GL(n)$, first developed by the speaker in mid-1990's.
The results are joint with Vladimir Shchigolev.

Abstract. Louis Kauffman found a special description of the Jones polynomial and the representation theory of $U_q(\mathfrak{sl}(2))$ in which each skein space has a basis of planar matchings. There is a similar calculus (discovered independently by myself and the late Francois Jaeger) for each of the three rank 2 simple Lie algebras $A_2$, $B_2$, and $G_2$. These skein theories, called "spiders", can also be viewed as Gröbner-type presentations of pivotal categories. In each of the four cases (optionally also including the semisimple case $A_1 \times A_1$), the Gröbner basis property yields a basis of skein diagrams called "webs". The basis webs are defined by an interesting non-positive curvature condition.
I will discuss a new connection between these spiders and the geometric Satake correspondence, which relates the representation category of a simple Lie algebra to an affine building of the Langlands dual algebra. In particular, any such building is ${\rm CAT}(0)$, which seems to explain the non-positive curvature of basis webs.

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Abstract. Khovanov-Lauda-Rouquier algebras have played a fundamental role in categorifying quantum groups. I will discuss the structure of their simple modules. This is joint work with Aaron Lauda.