Cohomology for quantum groups via the geometry of the nullcone

書誌事項

Cohomology for quantum groups via the geometry of the nullcone

Christopher P. Bendel ... [et.al.]

(Memoirs of the American Mathematical Society, no. 1077)

American Mathematical Society, 2014, c2013

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注記

Includes bibliographical references (p. 89-93)

"Volume 229, number 1077 (fourth of 5 numbers), May 2014"

Other authors: Daniel K. Nakano, Brian J. Parshall, Cornelius Pillen

内容説明・目次

内容説明

Let ζ be a complex ℓ th root of unity for an odd integer ℓ>1 . For any complex simple Lie algebra g , let u ζ =u ζ (g) be the associated "small" quantum enveloping algebra. This algebra is a finite dimensional Hopf algebra which can be realised as a subalgebra of the Lusztig (divided power) quantum enveloping algebra U ζ and as a quotient algebra of the De Concini-Kac quantum enveloping algebra U ζ . It plays an important role in the representation theories of both U ζ and U ζ in a way analogous to that played by the restricted enveloping algebra u of a reductive group G in positive characteristic p with respect to its distribution and enveloping algebras. In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when l (resp., p ) is smaller than the Coxeter number h of the underlying root system. For example, Lusztig's conjecture concerning the characters of the rational irreducible G -modules stipulates that p≥h . The main result in this paper provides a surprisingly uniform answer for the cohomology algebra H ∙ (u ζ ,C) of the small quantum group.

目次

Preliminaries and statement of results Quantum groups, actions, and cohomology Computation of Φ 0 and N(Φ 0 ) Combinatorics and the Steinberg module The cohomology algebra H ∙ (u ζ (g),C) Finite generation Comparison with positive characteristic Support varieties over u ζ for the modules ∇ ζ (λ) and Δ ζ (λ) Appendix A Bibliography

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