Cohomology for quantum groups via the geometry of the nullcone / Christopher P. Bendel [and three others].

Format:

Book

Author/Creator:

Bendel, Christopher P., 1969-

Bendel, Christopher P., 1969- author.

Contributor:

American Mathematical Society.

Series:

Memoirs of the American Mathematical Society ; Volume 229, Number 1077.

Memoirs of the American Mathematical Society, 1947-6221 ; Volume 229, Number 1077

Language:

English

Subjects (All):

Cohomology operations.

Algebraic topology.

Physical Description:

1 online resource (110 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2013.

Language Note:

English

Summary:

Let ζ be a complex ℓ th root of unity for an odd integer ℓ>1 . For any complex simple Lie algebra g , let u ζ =u ζ (g) be the associated "small" quantum enveloping algebra. This algebra is a finite dimensional Hopf algebra which can be realised as a subalgebra of the Lusztig (divided power) quantum enveloping algebra U ζ and as a quotient algebra of the De Concini-Kac quantum enveloping algebra U ζ . It plays an important role in the representation theories of both U ζ and U ζ 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≥h . The main result in this paper provides a surprisingly uniform answer for the cohomology algebra H ∙ (u ζ ,C) of the small quantum group.

Contents:

""Contents""; ""Introduction""; ""Chapter 1. Preliminaries and Statement of Results""; ""1.1. Some preliminary notation""; ""1.2. Main results""; ""Chapter 2. Quantum Groups, Actions, and Cohomology""; ""2.1. Listings""; ""2.2. Quantum enveloping algebras""; ""2.3. Connections with algebraic groups""; ""2.4. Root vectors and PBW-basis""; ""2.5. Levi and parabolic subalgebras""; ""2.6. The subalgebra _{ }( _{ })""; ""2.7. Adjoint action""; ""2.8. Finite dimensionality of cohomology groups""; ""2.9. Spectral sequences and the Euler characteristic""; ""2.10. Induction functors""

""Chapter 3. Computation of Î?â?€ and (Î?â?€)""""3.1. Subroot systems defined by weights""; ""3.2. The case of the classical Lie algebras""; ""3.3. The case of the exceptional Lie algebras""; ""3.4. Standardizing Î?â?€""; ""3.5. Resolution of singularities""; ""3.6. Normality of orbit closures""; ""Chapter 4. Combinatorics and the Steinberg Module""; ""4.1. Steinberg weights""; ""4.2. Weights of Î?^{â??}_{ , }""; ""4.3. Multiplicity of the Steinberg module""; ""4.4. Proof of Proposition 4.2.1""; ""4.5. The weight _{ }""; ""4.6. Types _{ }, _{ }, _{ }""; ""4.7. Type _{ }""

""4.8. Type _{ } with dividing +1""""4.9. Exceptional Lie algebras""; ""Chapter 5. The Cohomology Algebra ^{â??}( _{ }( ),â??)""; ""5.1. Spectral sequences, I""; ""5.2. Spectral sequences, II""; ""5.3. An identification theorem""; ""5.4. Spectral sequences, III""; ""5.5. Proof of main result, Theorem 1.2.3, I""; ""5.6. Spectral sequences, IV""; ""5.7. Proof of the main result, Theorem 1.2.3, II""; ""Chapter 6. Finite Generation""; ""6.1. A finite generation result""; ""6.2. Proof of part (a) of Theorem 1.2.4""; ""6.3. Proof of part (b) of Theorem 1.2.4""

""Chapter 7. Comparison with Positive Characteristic""""7.1. The setting""; ""7.2. Assumptions""; ""7.3. Consequences""; ""7.4. Special cases""; ""Chapter 8. Support Varieties over _{ } for the Modules â??_{ }( ) and Î?_{ }( )""; ""8.1. Quantum support varieties""; ""8.2. Lower bounds on the dimensions of support varieties""; ""8.3. Support varieties of â??_{ }( ): general results""; ""8.4. Support varieties of Î?_{ }( ) when is good""; ""8.5. A question of naturality of support varieties""; ""8.6. The Constrictor Method I""; ""8.7. The Constrictor Method II""

""8.8. Support varieties of â??_{ }( ) when is bad""""8.9. â?? when 3\mid ""; ""8.10. â?? when 3\mid ""; ""8.11. â?? when 3\mid ""; ""8.12. â?? when 3\mid , 5\mid ""; ""8.13. Support varieties of Î?_{ }( ) when is bad""; ""Appendix A.""; ""A.1. Tables I""; ""A.2. Tables II""; ""Bibliography""

Notes:

"Volume 229, Number 1077 (fourth of 5 numbers)."

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-1531-3