Ebook: Cohomology for quantum groups via the geometry of the nullcone
- Series: Memoirs of the American Mathematical Society 1077
- Year: 2014
- Publisher: American Mathematical Society
- Language: English
- pdf
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
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