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Cohomology for quantum groups via the geometry of the nullcone

About this Title

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

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 229, Number 1077
ISBNs: 978-0-8218-9175-9 (print); 978-1-4704-1531-0 (online)
Published electronically: October 16, 2013
Keywords:Quantum groups, cohomology

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Table of Contents


  • Introduction
  • Chapter 1. Preliminaries and Statement of Results
  • Chapter 2. Quantum Groups, Actions, and Cohomology
  • Chapter 3. Computation of $\Phi _{0}$ and ${\mathcal N}(\Phi _{0})$
  • Chapter 4. Combinatorics and the Steinberg Module
  • Chapter 5. The Cohomology Algebra $\operatorname {H}^{\bullet }(u_{\zeta }(\mathfrak {g}),\mathbb {C})$
  • Chapter 6. Finite Generation
  • Chapter 7. Comparison with Positive Characteristic
  • Chapter 8. Support Varieties over $u_{\zeta }$ for the Modules $\nabla _{\zeta }(\lambda )$ and $\Delta _{\zeta }(\lambda )$
  • Appendix A.


Let be a complex th root of unity for an odd integer . For any complex simple Lie algebra , let 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 and as a quotient algebra of the De Concini–Kac quantum enveloping algebra . It plays an important role in the representation theories of both and in a way analogous to that played by the restricted enveloping algebra of a reductive group in positive characteristic with respect to its distribution and enveloping algebras. In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when (resp., ) is smaller than the Coxeter number of the underlying root system. For example, Lusztig's conjecture concerning the characters of the rational irreducible -modules stipulates that . The main result in this paper provides a surprisingly uniform answer for the cohomology algebra of the small quantum group. When , this cohomology algebra has been calculated by Ginzburg and Kumar GK. Our result requires powerful tools from complex geometry and a detailed knowledge of the geometry of the nullcone of . In this way, the methods point out difficulties present in obtaining similar results for the restricted enveloping algebra in small characteristics, though they do provide some clarification of known results there also. Finally, we establish that if is a finite dimensional -module, then is a finitely generated -module, and we obtain new results on the theory of support varieties for .

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