Quantum current algebra $\mathbf{U}({\mathfrak{gl}}_n[t])$: canonical bases and polynomial representations

Abstract

We introduce a quantum deformation $\mathbf{U}({\mathfrak{gl}}_n[t])$ of the universal enveloping algebra of the current algebra ${\mathfrak{gl}}_n[t]$, realized as a Hopf parabolic subalgebra of quantum affine ${\mathfrak{gl}}_n$. Unlike the Yangian---the standard quantization of the current algebra---our algebra admits a canonical basis. We give a BLM-type realization of \(\mathbf{U}(\mathfrak{gl}_n[t])\) via certain subalgebras of affine quantum Schur algebras, and then construct canonical bases for the modified quantum current algebra $\dot{\mathbf{U}}({\mathfrak{gl}}_n[t])$ and for its finite dimensional irreducible graded modules. Moreover, we prove a rigidity theorem: every finite dimensional polynomial irreducible module for quantum affine ${\mathfrak{gl}}_n$ remains irreducible when restricted to ${\mathbf{U}}_{\mathbf{v}}({\mathfrak{gl}}_n[t])$ (the specialization of $\mathbf{U}({\mathfrak{gl}}_n[t])$ at a non-root-of-unity complex number $\mathbf{v}$); conversely, every finite dimensional affine polynomial irreducible ${\mathbf{U}}_{\mathbf{v}}({\mathfrak{gl}}_n[t])$-module extends uniquely to a polynomial irreducible module for quantum affine ${\mathfrak{gl}}_n$. More generally, we classify all finite dimensional polynomial irreducible \({\mathbf U}_{\mathbf v}(\frak{gl}_n[t])\)-modules by \(\mathcal Q(n)\timesΛ^+(n)\). As an application, we establish a bijection between the isomorphism classes of finite dimensional polynomial irreducible modules of \({\bf U}_{\mathbf v}(\frak{gl}_n[t])\) and those of the Yangian \(Y(\mathfrak{gl}_n)\). Thus \({\bf U}(\mathfrak{gl}_n[t])\) admits canonical bases for its modified form and irreducible graded modules, and its finite dimensional polynomial irreducible modules are in correspondence with those of the Yangian.

arXiv comment49 pages