Constructing quantum queer supergroups using Hecke-Clifford superalgebras

Abstract

There are two important classes of H-algebras --- Hecke algebras and (Ringel-)Hall algebras, which provide both geometric and algebraic approaches to quantum groups. In early 90s, only quantum linear groups can be constructed using Hecke algebras of symmetric groups, while the Hall algebras associated with Dynkin quivers with automorphisms provide a realization for the positive part of the corresponding quantum groups.  We recorded the two beautiful theories in book form as part of mutually enriching interactions between ring theory and Lie theory. 

Almost twenty years after the book publication, both theories have advanced significantly. The Hecke algebra approach has been extended to affine type A, to super type A, and to finite types B/C/D, while the Hall algebra approach has been extended to the entire quantum groups, to i-quantum groups, and to some affine cases.

In the first talk, I will focus on the Hecke algebra approach through their associated q-Schur algebras of classical types and show how to use certain multiplication formulas in q-Schur algebras to construct quantum/i-quantum groups and their canonical basis theory.  

For the second talk: Using a geometric setting of q-Schur algebras, Beilinson-Lusztig-MacPherson discovered a new basis for quantum gl_n (i.e., the quantum enveloping algebra Uq(gl_n) of the Lie algebra gl_n) and its associated matrix representation of the regular module of Uq(gl_n). This beautiful work shows that the structure of the quantum linear group is hidden in the structure of Hecke algebras. The work has been generalized (either geometrically or algebraically) to quantum affine gl_n, quantum super gl_{m|n}, and recently, to some i-quantum groups of type AIII. In this talk, I will report on a completion of the work for a new construction of the quantum queer supergroup using Hecke-Clifford superalgebras and their associated queer q-Schur superalgberas.