Abstract:

These lectures will explore connections between (generalized) q-Schur algebras and the quantized enveloping algebra Uq(g) associated with a simple Lie algebra g. These connections are facilitated by a certain completion of Lusztig’s modified form of Uq(g). Although the q-Schur algebras arose initially as quotients of Uq(g) it is possible to reverse history and use them as a tool to reconstruct Uq(g). In type A, q-Schur algebras arise naturally in connection with a q-analogue of Schur–Weyl duality discovered by Jimbo (1986) which links the representation theories of Uq(gl_n) and the Iwahori–Hecke algebra Hq(S_r) of the symmetric group S_r.