Cohomology for quantum groups via the geometry of the nullcone
Author(s)
Bibliographic Information
Cohomology for quantum groups via the geometry of the nullcone
（Memoirs of the American Mathematical Society, no. 1077）
American Mathematical Society, 2014, c2013
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Note
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
Description and Table of Contents
Description
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.
Table of Contents
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
by "Nielsen BookData"