Cohomology for Quantum Groups Via the Geometry of the Nullcone(English, Paperback, Bendel Christopher P.)

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Cohomology for Quantum Groups Via the Geometry of the Nullcone(English, Paperback, Bendel Christopher P.) is written by Bendel Christopher P. and published by American Mathematical Society. It's available with International Standard Book Number or ISBN identification 0821891758 (ISBN 10) and 9780821891759 (ISBN 13).

Let ? be a complex ? th root of unity for an odd integer ?>1 . For any complex simple Lie algebra g , let u ? =u ? (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 ? and as a quotient algebra of the De Concini-Kac quantum enveloping algebra U ? . It plays an important role in the representation theories of both U ? and U ? 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 ? ,C) of the small quantum group.