The Hopf algebra Rep $ U_{q} \widehat{\frak g \frak l}_\infty $

Abstract

We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of \( U_{q} \widehat{\frak g \frak l}_N \) in the limit \( N \to \infty \). The resulting Hopf algebra Rep \( U_{q} \widehat{\frak g \frak l}_\infty \) is a tensor product of its Hopf subalgebras \( \textrm{Rep}_{a} U_{q} \widehat{\frak g \frak l}_\infty \), \( a \in \mathbb{C}^\times/q^{2{\mathbb Z}} \). When q is generic (resp., q2 is a primitive root of unity of order l), we construct an isomorphism between the Hopf algebra \( \textrm{Rep}_{a} U_{q} \widehat{\frak g \frak l}_\infty \) and the algebra of regular functions on the prounipotent proalgebraic group \( \widetilde{SL}_{\infty}^{-} \) (resp., \( \widetilde{GL}_{l}^{-} \)). When q is a root of unity, this isomorphism identifies the Hopf subalgebra of \( \textrm{Rep}_{a} U_{q} \widehat{\frak g \frak l}_\infty \) 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 \( \widetilde{GL}_{l}^{-} \) considered as an \( l \times l \) matrix over the Taylor series. 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 \( \textrm{Rep}_{a} U_{q} \widehat{\frak g \frak l}_\infty \) and describe the span of tensor products of evaluation representations taken at fixed points as a module over this Hall algebra.