Cohomology for Quantum Groups Via the Geometry of the Nullcone

Christopher P. Bendel, Daniel K. Nakano, Brian J. Parshall, Cornelius Pillen (Autoren)

Buch | Softcover

93 Seiten

2014
American Mathematical Society (Verlag)
978-0-8218-9175-9 (ISBN)

Titel z.Zt. nicht lieferbar
Versandkostenfrei
Auch auf Rechnung

Artikel merken

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.

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.

Preliminaries and statement of results
Quantum groups, actions, and cohomology
Computation of Φ 0 and N(Φ 0 )
Combinatorics and the Steinberg module
The cohomology algebra H ∙ (u ζ (g),C)
Finite generation
Comparison with positive characteristic
Support varieties over u ζ for the modules ∇ ζ (λ) and Δ ζ (λ)
Appendix A
Bibliography

Erscheint lt. Verlag	22.9.2014
Reihe/Serie	Memoirs of the American Mathematical Society
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	172 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	0-8218-9175-8 / 0821891758
ISBN-13	978-0-8218-9175-9 / 9780821891759
Zustand	Neuware