Quantum ${\mathfrak{gl}}_{\infty }$, infinite $q$-Schur algebras and their representations

Abstract

In this paper, we investigate the structure and representations of the quantum group $\mathbf{U}(\infty )={\mathbf{U}}_{\upsilon }({\mathfrak{gl}}_{\infty })$. We will present a realization for $\mathbf{U}(\infty )$, following Beilinson--Lusztig--MacPherson (BLM) \cite{BLM}, and show that the natural algebra homomorphism $\hat{I}{\P }_r$ from $\mathbf{U}(\infty )$ to the infinite $q$-Schur algebra $\mathcal{S}(\infty ,r)$ is not surjective for any $r\ge 1$. We will give a BLM type realization for the image $\mathbf{U}(\infty ,r):=\hat{I}{\P }_r(\mathbf{U}(\infty ))$ and discuss its presentation in terms of generators and relations. We further construct a certain completion algebra ${\hat{\mathcal{K}}}^{†}(\infty )$ so that $\hat{I}{\P }_r$ can be extended to an algebra epimorphism $\tilde{\hat{I}}{\P }_r:{\hat{\mathcal{K}}}^{†}(\infty )\to \mathcal{S}(\infty ,r)$. Finally we will investigate the representation theory of $\mathbf{U}(\infty )$, especially the polynomial representations of $\mathbf{U}(\infty )$.