The Hopf algebra $Rep U_q \hat{gl}_\infty$

Abstract

We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of $U_q \hat{gl}_N$ in the limit $N \to \infty$. The resulting Hopf algebra $Rep U_q \hat{gl}_\infty$ is a tensor product of its Hopf subalgebras $Rep_a U_q \hat{gl}_\infty$, $a\in\C^\times/q^{2\Z}$. When $q$ is generic (resp., $q^2$ is a primitive root of unity of order $l$), we construct an isomorphism between the Hopf algebra $Rep_a U_q \hat{gl}_\infty$ and the algebra of regular functions on the prounipotent proalgebraic group $SL_\infty^-$ (resp., $\tilde{GL}_l^-$). When $q$ is a root of unity, this isomorphism identifies the Hopf subalgebra of $Rep_a U_q \hat{gl}_\infty$ spanned by the modules obtained by pullback with respect to the Frobenius homomorphism with the algebra generated by the coefficients of the determinant of an element of $\tilde{GL}_l^-$. This gives us an explicit formula for the Frobenius pullbacks of the fundamental representations. In addition, we construct a natural action of the Hall algebra associated to the infinite linear quiver (resp., the cyclic quiver with $l$ vertices) on $Rep_a U_q \hat{gl}_infty$ and describe the span of the tensor products of the evaluation representations taken at fixed points as a module over this Hall algebra.