In this paper we compute extension groups in the category of strict
polynomial superfunctors and thereby exhibit certain "universal extension
classes" for the general linear supergroup. Some of these classes restrict to
the universal extension classes for the general linear group exhibited by
Friedlander and Suslin, while others arise from purely super phenomena. We then
use these extension classes to show that the cohomology ring of a finite
supergroup scheme---equivalently of a finite-dimensional cocommutative Hopf
superalgebra---over a field is a finitely-generated algebra. Implications for
the rational cohomology of the general linear supergroup are also discussed.Comment: Corrected minor typos and added additional details to some proof
Abstract. We prove that the cohomology ring of a finite-dimensional restricted Lie superalgebra over a field of characteristic p > 2 is a finitelygenerated algebra. Our proof makes essential use of the explicit projective resolution of the trivial module constructed by J. Peter May for any graded restricted Lie algebra. We then prove that the cohomological finite generation problem for finite supergroup schemes over fields of odd characteristic reduces to the existence of certain conjectured universal extension classes for the general linear supergroup GL(m|n) that are similar to the universal extension classes for GL n exhibited by Friedlander and Suslin.
a b s t r a c tLet U ζ be the quantum group (Lusztig form) associated to the simple Lie algebra g, with parameter ζ specialized to an ℓ-th root of unity in a field of characteristic p > 0. In this paper we study certain finite-dimensional normal Hopf subalgebras U ζ (G r ) of U ζ , called Frobenius-Lusztig kernels, which generalize the Frobenius kernels G r 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 U ζ (G r )-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.
The proof of Theorem 5.12 in [1] does not make sense as written because the algebra u ζ (b + α ) need not be a Hopf subalgebra of u ζ (b + ) unless α is a simple root. This note describes how the proof should be modified to work around this fact.All notation is taken from [1]. Theorem 5.12 of [1] is as follows:The proof given in [1] involves considering the restriction of M to a certain subalgebra u ζ (b + α ) of u ζ (b + ). However, the proof does not make sense as written because, while u ζ (b + α ) does admit the structure of a Hopf algebra, it need not be a Hopf subalgebra of u ζ (b + ) unless α is a simple root. The purpose of this note is to give a correct proof of the theorem.Proof of Theorem 5.12. Let α ∈ Φ + be a positive root. Write α = β∈Π m β β as a sum of simple roots, and set K α = β∈Π K
The authors compute the support varieties of all irreducible modules for the small quantum group u ζ (g), where g is a simple complex Lie algebra, and ζ is a primitive ℓ-th root of unity with ℓ larger than the Coxeter number of g. The calculation employs the prior calculations and techniques of Ostrik and of Nakano-Parshall-Vella, as well as deep results involving the validity of the Lusztig character formula for quantum groups and the positivity of parabolic Kazhdan-Lusztig polynomials for the affine Weyl group. Analogous support variety calculations are provided for the first Frobenius kernel G1 of a reductive algebraic group scheme G defined over the prime field Fp.
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