Frenkel And
نویسنده
چکیده
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 functions 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 pullback with respect to the Frobenius homomorphism with the algebra generated by the coefficients of the determinant of an element of G̃L − l considered as an l× 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 Repa Uq ĝl∞ and describe the span of tensor products of evaluation representations taken at fixed points as a module over this Hall algebra.
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