A MODIFIED BLM APPROACH TO QUANTUM AFFINE gln

We introduce a spanning set of BLM type, {A(j, r)}A,j, for affine quantum Schur algebras S△(n, r) and construct a linearly independent set {A(j)}A,j for an associated algebra K̂△(n). We then establish explicitly some multiplication formulas of simple generators E △ h,h+1(0) by an arbitrary element A(j) in K̂△(n) via the corresponding formulas in S△(n, r), and compare these formulas with the multiplication formulas between a simple module and an arbitrary module in the Ringel–Hall algebras H△(n) associated with cyclic quivers. This allows us to use the triangular relation between monomial and PBW type bases for H△(n) established in [2] to derive similar triangular relations for S△(n, r) and K̂△(n). Using these relations, we then show that the subspace A△(n) of K̂△(n) spanned by {A(j)}A,j contains the quantum enveloping algebra U△(n) of affine type A as a subalgebra. As an application, we prove that, when this construction is applied to quantum Schur algebras S(n, r), the resulting subspace A(n) is in fact a subalgebra which is isomorphic to the quantum enveloping algebra of gln. We conjecture that A△(n) is a subalgebra.