The Integral quantum loop algebra of ${\mathfrak{gl}}_n$

Abstract

We will construct the Lusztig form for the quantum loop algebra of ${\mathfrak{gl}}_n$ by proving the conjecture \cite[3.8.6]{DDF} and establish partially the Schur--Weyl duality at the integral level in this case. We will also investigate the integral form of the modified quantum affine ${\mathfrak{gl}}_n$ by introducing an affine stabilisation property and will lift the canonical bases from affine quantum Schur algebras to a canonical basis for this integral form. As an application of our theory, we will also discuss the integral form of the modified extended quantum affine ${\mathfrak{sl}}_n$ and construct its canonical basis to verify a conjecture of Lusztig in this case.

30 Pages