Quantum current algebra $\mathbf{U}({\mathfrak{gl}}_n[t])$: canonical bases, rigidity, and relation with Yangians

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 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 polynomial irreducible ${\mathbf{U}}_{\mathbf{v}}({\mathfrak{gl}}_n[t])$-module extends uniquely to a polynomial irreducible module for quantum affine ${\mathfrak{gl}}_n$. Consequently, the finite dimensional polynomial irreducible modules of ${\mathbf{U}}_{\mathbf{v}}({\mathfrak{gl}}_n[t])$ are in bijection with those of the Yangian $Y({\mathfrak{gl}}_n)$. This provides the first example of a quantum current algebra with a well-developed canonical basis theory, providing new combinatorial approaches to the representation theory of current algebras.

47 pages