Cohomology for quantum groups via the geometry of the nullcone

Bendel, Christopher; Nakano, Daniel; Parshall, Brian; Pillen, Cornelius

doi:10.1090/memo/1077

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)
DOI: http://dx.doi.org/10.1090/memo/1077
Published electronically: October 16, 2013
Keywords:Quantum groups, cohomology

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Chapters

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.

Abstract

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. When $\ell >h$ , 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 $\mathfrak g$ . In this way, the methods point out difficulties present in obtaining similar results for the restricted enveloping algebra $u$ in small characteristics, though they do provide some clarification of known results there also. Finally, we establish that if $M$ is a finite dimensional $u_\zeta$ -module, then $\operatorname {H}^\bullet (u_\zeta ,M)$ is a finitely generated $\operatorname {H}^\bullet (u_\zeta ,\mathbb C)$ -module, and we obtain new results on the theory of support varieties for $u_\zeta$ .

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Cohomology for quantum groups via the geometry of the nullcone