2 Edward Frenkel and Evgeny Mukhin

We define the Hopf algebra structure on the Grothendieck group of finitedimensional polynomial representations of Uq ĝlN in the limit N → ∞. The resulting Hopf algebra Rep Uq ĝl∞ is a tensor product of its Hopf subalgebras Repa Uqĝl∞, a ∈ C/q. When q is generic (resp., q is a primitive root of unity of order l), we construct an isomorphism between the Hopf algebra Repa Uq ĝl∞ and the algebra of regular funtions on the prounipotent proalgebraic group S̃L − ∞ (resp., G̃L − l ). When q is a root of unity, this isomorphism identifies the Hopf subalgebra of Repa Uq ĝl∞ spanned by the modules obtained by pull-back with respect to the Frobenius homomorphism, with the algebra of functions on the center of G̃L − l . 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 Repa Uqĝl∞ and describe the span of the tensor products of evaluation representations taken at fixed points as a module over this Hall algebra.