## University of Sydney Algebra Seminar

# Jie Du (University of New South Wales)

## Friday 5th June, 12.05-12.55pm, Carslaw 375

### A modified BLM approach to quantum affine gl_n

We introduce BLM type bases for affine quantum Schur algebras. We then establish certain multiplication formulas in these algebras, and compare these formulas with the multiplication formulas between a simple module and an arbitrary module in the Ringel-Hall algebras associated with cyclic quivers. This allows us to use the triangular relation between monomial and PBW bases for the Ringel-Hall algebra to derive similar triangular relations for affine quantum Schur algebras. Using these relations, we then show that the quantum enveloping algebra of affine type $A$ can be embedded into the algebra obtained by taking a "completion" of the direct sum algebra of all affine quantum Schur algebras. Possible applications to double Hall algebras and connections with various existing works by Lusztig, Schiffmann, Hubery, Varaganolo-Vasserot, McGerty ... will also be briefly discussed.

This is joint work with Qiang Fu.