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.
Categories: math.QA math.RT