A modified BLM approach to quantum affine ${\mathfrak{gl}_n}$

Abstract

We introduce a spanning set of Beilinson–Lusztig–MacPherson type, {A(j, r)}_A,j, for affine quantum Schur algebras ${{{\boldsymbol{\mathcal S}}_\vartriangle}(n, r)}$ and construct a linearly independent set {A(j)}_A,j for an associated algebra ${{{\boldsymbol{\widehat{\mathcal K}}}_\vartriangle}(n)}$ . We then establish explicitly some multiplication formulas of simple generators ${E^\vartriangle_{h,h+1}}(\mathbf{0})$ by an arbitrary element A(j) in ${{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle(n)}}$ via the corresponding formulas in ${{{\boldsymbol{\mathcal S}}_\vartriangle(n, r)}}$ , and compare these formulas with the multiplication formulas between a simple module and an arbitrary module in the Ringel–Hall algebras ${{{\boldsymbol{\mathfrak H}_\vartriangle(n)}}}$ associated with cyclic quivers. This allows us to use the triangular relation between monomial and PBW type bases for ${{\boldsymbol{\mathfrak H}}_\vartriangle}(n)$ established in Deng and Du (Adv Math 191:276–304, 2005) to derive similar triangular relations for ${{{\boldsymbol{\mathcal S}}_\vartriangle}(n, r)}$ and ${{\boldsymbol{\widehat{\mathcal K}}}_\vartriangle}(n)$ . Using these relations, we then show that the subspace ${{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)}$ of ${{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle}(n)}$ spanned by {A(j)}_A,j contains the quantum enveloping algebra ${{{\mathbf U}_\vartriangle}(n)}$ of affine type A as a subalgebra. As an application, we prove that, when this construction is applied to quantum Schur algebras ${\boldsymbol{\mathcal S}(n,r)}$ , the resulting subspace ${{{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)}}$ is in fact a subalgebra which is isomorphic to the quantum enveloping algebra of ${\mathfrak{gl}_n}$ . We conjecture that ${{{{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)}}}$ is a subalgebra of ${{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle}(n)}$ .