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Cohomology for Quantum Groups via the Geometry of the Nullcone
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, and Cornelius Pillen, University of South Alabama, Mobile, Alabama

Memoirs of the American Mathematical Society
2013; 93 pp; softcover
Volume: 229
ISBN-10: 0-8218-9175-8
ISBN-13: 978-0-8218-9175-9
List Price: US$71
Individual Members: US$42.60
Institutional Members: US$56.80
Order Code: MEMO/229/1077
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Let \(\zeta\) be a complex \(\ell\)th root of unity for an odd integer \(\ell>1\). For any complex simple Lie algebra \(\mathfrak g\), let \(u_\zeta=u_\zeta({\mathfrak g})\) be the associated "small" quantum enveloping algebra. This algebra is a finite dimensional Hopf algebra which can be realized as a subalgebra of the Lusztig (divided power) quantum enveloping algebra \(U_\zeta\) and as a quotient algebra of the De Concini-Kac quantum enveloping algebra \({\mathcal U}_\zeta\). It plays an important role in the representation theories of both \(U_\zeta\) and \({\mathcal U}_\zeta\) 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 \(p \geq h\). The main result in this paper provides a surprisingly uniform answer for the cohomology algebra \(\operatorname{H}^\bullet(u_\zeta,{\mathbb C})\) of the small quantum group.

Table of Contents

  • Preliminaries and statement of results
  • Quantum groups, actions, and cohomology
  • Computation of \(\Phi_{0}\) and \({\mathcal N}(\Phi_{0})\)
  • Combinatorics and the Steinberg module
  • The cohomology algebra \(\operatorname{H}^{\bullet}(u_{\zeta}(\mathfrak{g}),\mathbb{C})\)
  • Finite generation
  • Comparison with positive characteristic
  • Support varieties over \(u_{\zeta}\) for the modules \(\nabla_{\zeta}(\lambda)\) and \(\Delta_{\zeta}(\lambda)\)
  • Appendix A
  • Bibliography
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