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Cohomology for Quantum Groups via the Geometry of the Nullcone
Christopher P. BendelDaniel K. Nakano Brian J. ParshalCornelius Pillen
出版American Mathematical Soc., 2014-04-07
主題Mathematics / Algebra / GeneralMathematics / Algebra / LinearMathematics / Geometry / AlgebraicMathematics / Topology
ISBN08218917589780821891759
URLhttp://books.google.com.hk/books?id=-ANJAwAAQBAJ&hl=&source=gbs_api
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註釋Let ζ be a complex lth root of unity for an odd integer l>1. For any complex simple Lie algebra g, let uζ=uζ(g) be the associated “small” quantum enveloping algebra. This algebra is a finite dimensional Hopf algebra which can be realized as a subalgebra of the Lusztig (divided power) quantum enveloping algebra Uζ and as a quotient algebra of the De Concini–Kac quantum enveloping algebra Uζ. It plays an important role in the representation theories of both Uζ and Uζ 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∙⁡(uζ,C) of the small quantum group. When l > h, this cohomology algebra has been calculated by Ginzburg and Kumar GK. Our result requires powerful tools from complex geometry and a detailed knowledge of the geometry of the nullcone of g. In this way, the methods point out difficulties present in obtaining similar results for the restricted enveloping algebra u u in small characteristics, though they do provide some clarification of known results there also. Finally, we establish that if M is a finite dimensional uζ-module, then H∙⁡(uζ,M) is a finitely generated H∙⁡(uζ,C)-module, and we obtain new results on the theory of support varieties for uζ.