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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
Cornelius Pillen University of South Alabama, Mobile, Alabama
Cohomology for Quantum Groups via the Geometry of the Nullcone
eBook ISBN:  978-1-4704-1531-0
Product Code:  MEMO/229/1077.E
List Price: $71.00
MAA Member Price: $63.90
AMS Member Price: $42.60
Cohomology for Quantum Groups via the Geometry of the Nullcone
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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
Cornelius Pillen University of South Alabama, Mobile, Alabama
eBook ISBN:  978-1-4704-1531-0
Product Code:  MEMO/229/1077.E
List Price: $71.00
MAA Member Price: $63.90
AMS Member Price: $42.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2292013; 93 pp
    MSC: Primary 20; Secondary 17;

    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
     
     
    • Chapters
    • Introduction
    • 1. Preliminaries and Statement of Results
    • 2. Quantum Groups, Actions, and Cohomology
    • 3. Computation of $\Phi _{0}$ and ${\mathcal N}(\Phi _{0})$
    • 4. Combinatorics and the Steinberg Module
    • 5. The Cohomology Algebra $\operatorname {H}^{\bullet }(u_{\zeta }(\mathfrak {g}),\mathbb {C})$
    • 6. Finite Generation
    • 7. Comparison with Positive Characteristic
    • 8. Support Varieties over $u_{\zeta }$ for the Modules $\nabla _{\zeta }(\lambda )$ and $\Delta _{\zeta }(\lambda )$
    • Appendix A.
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2292013; 93 pp
MSC: Primary 20; Secondary 17;

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.

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