Cohomology for Quantum Groups via the Geometry of the Nullcone (Memoirs of the American Mathematical Society) - Softcover

Bendel, Christopher P.; Nakano, Daniel K.; Parshall, Brian J.; Pillen, Cornelius

Softcover

ISBN 10: 0821891758 ISBN 13: 9780821891759

Publisher: American Mathematical Society, 2014

Synopsis

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.

About this title

About the Author

Christopher P. Bendel, University of Wisconsin-Stout, Menomonie, Wisconsin.

Daniel K. Nakano, University of Georgia, Athens.

Georgia, Brian J. Parshall, University of Virginia, Charlottesville, Virginia.

Cornelius Pillen, University of South Alabama, Mobile, Alabama.

Edition details

• Publisher: American Mathematical Society
• Publication date: 2014
• Language: English
• ISBN 10: 0821891758
• ISBN 13: 9780821891759
• Binding: Paperback
• Number of pages: 93