Another realization of the category of modules over the small quantum group

By Sergey V. Arkhipov, D. Gaitsgory. 2000

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Abstract Let g be a semi-simple simply-connected Lie algebra and let Ul be the corresponding quantum group with divided powers, where l is an even order root of unity. Let in addition ul⊂Ul be the corresponding “small” quantum group. In this paper we establish the following relation between the categories of representations of Ul and ul. We show that the category of ul-modules is naturally equivalent to the category of Ul-modules, which have a Hecke eigen-property with respect to representations lifted by means of the quantum Frobenius map U l →U( g ) , where g is the Langlands dual Lie algebra. This description allows to express the regular linkage class in the category ul-mod in terms of perverse sheaves on the affine flag variety with a Hecke eigen-property. Moreover, it can serve as a basis to the program to understand the connection between the category ul-mod and the category of representations of the corresponding affine algebra at the critical level.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/aa00d0097d7ef6410126571f807ee4bb238138ff

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Citations in the corpus

Non-group-theoretical semisimple Hopf algebras from group actions on fusion categories (2007, methodology): “The construction of these categories involves an equivariantization procedure [25, 1, 9]: given an action of a finite group G on a tensor category C one forms a tensor category C of G-equivariant objects in C.”
Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves (2004, methodology): “F r ∗ (V ∗ λ )) where Vλ is an irreducible U(g) module with highest weight λ (see [ AG ]).”
On braided fusion categories I (2009, background): “8, we recall the following well-known principle (see [2,20,21]): the datum of a Karoubian k-linear category with an action of G is equivalent to the datum of a Karoubian k-linear category with a k-linear action of Rep(G).”
Exact sequences of tensor categories (2010, background): “The G-equivariantization of C, denoted C, is, by definition, the category of G-equivariant objects and G-equivariant morphisms [2, 17, 32, 35].”