# Cohomology for Quantum Groups via the Geometry of the Nullcone

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*Christopher P. Bendel; Daniel K. Nakano; Brian J. Parshall; Cornelius Pillen*

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

# Table of Contents

## Cohomology for Quantum Groups via the Geometry of the Nullcone

- Introduction vii8 free
- Chapter 1. Preliminaries and Statement of Results 112 free
- Chapter 2. Quantum Groups, Actions, and Cohomology 718
- 2.1. Listings 718
- 2.2. Quantum enveloping algebras 819
- 2.3. Connections with algebraic groups 1021
- 2.4. Root vectors and PBW-basis 1122
- 2.5. Levi and parabolic subalgebras 1223
- 2.6. The subalgebra 𝕌_{𝕢}(𝔲_{𝔍}) 1223
- 2.7. Adjoint action 1324
- 2.8. Finite dimensionality of cohomology groups 1627
- 2.9. Spectral sequences and the Euler characteristic 1728
- 2.10. Induction functors 2031

- Chapter 3. Computation of Φ₀ and 𝒩(Φ₀) 2132
- Chapter 4. Combinatorics and the Steinberg Module 3142
- 4.1. Steinberg weights 3142
- 4.2. Weights of Λ^{∙}_{𝜁,𝐽} 3243
- 4.3. Multiplicity of the Steinberg module 3344
- 4.4. Proof of Proposition 4.2.1 3445
- 4.5. The weight 𝛿_{𝑋} 3546
- 4.6. Types 𝐵_{𝑛},𝐶_{𝑛},𝐷_{𝑛} 3849
- 4.7. Type 𝐴_{𝑛} 3950
- 4.8. Type 𝐴_{𝑛} with 𝑙 dividing 𝑛+1 4152
- 4.9. Exceptional Lie algebras 4354

- Chapter 5. The Cohomology Algebra 𝐻^{∙}(𝑢_{𝜁}(𝔤),ℂ) 4960
- Chapter 6. Finite Generation 6172
- Chapter 7. Comparison with Positive Characteristic 6576
- Chapter 8. Support Varieties over 𝑢_{𝜁} for the Modules ∇_{𝜁}(𝜆) and Δ_{𝜁}(𝜆) 7182
- 8.1. Quantum support varieties 7182
- 8.2. Lower bounds on the dimensions of support varieties 7182
- 8.3. Support varieties of ∇_{𝜁}(𝜆): general results 7283
- 8.4. Support varieties of Δ_{𝜁}(𝜆) when 𝑙 is good 7384
- 8.5. A question of naturality of support varieties 7485
- 8.6. The Constrictor Method I 7586
- 8.7. The Constrictor Method II 7586
- 8.8. Support varieties of ∇_{𝜁}(𝜆) when 𝑙 is bad 7687
- 8.9. 𝐸₆ when 3\mid𝑙 7788
- 8.10. 𝐹₄ when 3\mid𝑙 7788
- 8.11. 𝐸₇ when 3\mid𝑙 7889
- 8.12. 𝐸₈ when 3\mid𝑙, 5\mid𝑙 7990
- 8.13. Support varieties of Δ_{𝜁}(𝜆) when 𝑙 is bad 8091

- Appendix A. 8192
- Bibliography 89100