Cohomology for Quantum Groups Via the Geometry of the Nullcone
Christopher P. Bendel, Daniel K. Nakano, Brian J. Parshall, "Cohomology for Quantum Groups Via the Geometry of the Nullcone"
English | 2014 | pages: 110 | ISBN: 0821891758 | PDF | 1,0 mb
English | 2014 | pages: 110 | ISBN: 0821891758 | PDF | 1,0 mb
Let ζ be a complex ℓ th root of unity for an odd integer ℓ1 . 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.
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