The Integral Quantum Loop Algebra of $\mathfrak {gl}_{n}$

Abstract

Abstract We will construct the Lusztig form for the quantum loop algebra of $\mathfrak {gl}_{n}$ by proving the conjecture [4, 3.8.6] 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 provide an alternative algebra structure related to a conjecture of Lusztig in [29, §9.3], which has been already proved in [34].