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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
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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

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.

• Computation of $$\Phi_{0}$$ and $${\mathcal N}(\Phi_{0})$$
• The cohomology algebra $$\operatorname{H}^{\bullet}(u_{\zeta}(\mathfrak{g}),\mathbb{C})$$
• Support varieties over $$u_{\zeta}$$ for the modules $$\nabla_{\zeta}(\lambda)$$ and $$\Delta_{\zeta}(\lambda)$$