The Hopf algebra Rep Uq gl∞

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We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of Uq glₙ in the limit N → ∞. The resulting Hopf algebra Rep Uq gl∞ is a tensor product of its Hopf subalgebras Repₐ Uq gl∞, a∈×/q². When q is generic (resp., q² is a primitive root of unity of order l), we construct an isomorphism between the Hopf algebra Repₐ Uq gl∞ and the algebra of regular functions on the prounipotent proalgebraic group SL∞- (resp., GLₗ-). When q is a root of unity, this isomorphism identifies the Hopf subalgebra of Repₐ Uq gl∞ 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 GLₗ-. 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 Repₐ Uq glᵢnfty and describe the span of the tensor products of the evaluation representations taken at fixed points as a module over this Hall algebra.