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.

arXiv comment30 Pages

References in corpus (3)

• BLM realization for ${\mathcal{U}}_{\mathbb{Z}}({\widehat{\mathfrak{gl}}}_n)$
• Quantum affine ${\mathfrak{gl}}_n$ via Hecke algebras
• BLM realization for the integral form of quantum ${\mathfrak{gl}}_n$

Cited by in corpus (8)

• Geometric Schur duality of classical type
• Affine flag varieties and quantum symmetric pairs
• An elementary construction of monomial bases of modified quantum affine ${\mathfrak{gl}}_n$
• Positivity properties for canonical bases of modified quantum affine ${\mathfrak{sl}}_n$
• On the hyperalgebra of the loop algebra ${\widehat{\mathfrak{gl}}}_n$
• Presenting affine Schur Algebras
• The comultiplication of modified quantum affine ${\mathfrak{sl}}_n$
• Remarks on PBW bases of Ringel-Hall algebras of cyclic quivers