Representations and cohomology for Frobenius-Lusztig kernels |
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Authors: | Christopher M Drupieski |
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Institution: | Department of Mathematics, University of Georgia, Athens, GA 30602-7403, United States |
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Abstract: | Let Uζ be the quantum group (Lusztig form) associated to the simple Lie algebra g, with parameter ζ specialized to an ?-th root of unity in a field of characteristic p>0. In this paper we study certain finite-dimensional normal Hopf subalgebras Uζ(Gr) of Uζ, called Frobenius-Lusztig kernels, which generalize the Frobenius kernels Gr of an algebraic group G. When r=0, the algebras studied here reduce to the small quantum group introduced by Lusztig. We classify the irreducible Uζ(Gr)-modules and discuss their characters. We then study the cohomology rings for the Frobenius-Lusztig kernels and for certain nilpotent and Borel subalgebras corresponding to unipotent and Borel subgroups of G. We prove that the cohomology ring for the first Frobenius-Lusztig kernel is finitely-generated when g has type A or D, and that the cohomology rings for the nilpotent and Borel subalgebras are finitely-generated in general. |
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Keywords: | 20G42 20G05 20G10 |
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