The hopf algebra rep$U_q \widehat{\mathfrak{g}\mathfrak{l}}_\infty$

Abstract

We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of$U_q \widehat{\mathfrak{g}\mathfrak{l}}_N$ in the limitN→∞. The resulting Hopf algebra Rep$U_q \widehat{\mathfrak{g}\mathfrak{l}}_\infty$ is a tensor product of its Hopf subalgebras Rep_a $U_q \widehat{\mathfrak{g}\mathfrak{l}}_\infty$,a ∈ ℂ^×/q^2ℤ. Whenq is generic (resp.,q ² is a primitive root of unity of orderl), we construct an isomorphism between the Hopf algebra Rep _a $U_q \widehat{\mathfrak{g}\mathfrak{l}}_\infty$ and the algebra of regular functions on the prounipotent proalgebraic group$\widetilde{SL}\overline {_\infty }$ (resp.,$\widetilde{GL}\overline {_l }$). Whenq is a root of unity, this isomorphism identifies the Hopf subalgebra of Rep _a $U_q \widehat{\mathfrak{g}\mathfrak{l}}_\infty$ 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$\widetilde{GL}\overline {_l }$ considered as anl×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 withl vertices) on Rep _a $U_q \widehat{\mathfrak{g}\mathfrak{l}}_\infty$ and describe the span of tensor products of evaluation representations taken at fixed points as a module over this Hall algebra.