**Memoirs of the American Mathematical Society**

2013;
91 pp;
Softcover

MSC: Primary 16;
Secondary 20; 17; 53

Print ISBN: 978-0-8218-9174-2

Product Code: MEMO/229/1078

List Price: $71.00

AMS Member Price: $42.60

MAA Member Price: $63.90

**Electronic ISBN: 978-1-4704-1532-7
Product Code: MEMO/229/1078.E**

List Price: $71.00

AMS Member Price: $42.60

MAA Member Price: $63.90

# On the Spectra of Quantum Groups

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*Milen Yakimov*

Joseph and Hodges–Levasseur (in the A case) described the spectra of all quantum function algebras \(R_q[G]\) on simple algebraic groups in terms of the centers of certain localizations of quotients of \(R_q[G]\) by torus invariant prime ideals, or equivalently in terms of orbits of finite groups. These centers were only known up to finite extensions. The author determines the centers explicitly under the general conditions that the deformation parameter is not a root of unity and without any restriction on the characteristic of the ground field. From it he deduces a more explicit description of all prime ideals of \(R_q[G]\) than the previously known ones and an explicit parametrization of \(\mathrm{Spec} R_q[G]\).

#### Table of Contents

# Table of Contents

## On the Spectra of Quantum Groups

- Chapter 1. Introduction 18 free
- Chapter 2. Previous results on spectra of quantum function algebras 916 free
- 2.1. Quantized universal enveloping algebras 916
- 2.2. Type 1 modules and braid group action 1017
- 2.3. 𝐻-prime ideals of Quantum Groups 1118
- 2.4. Sets of normal elements 1219
- 2.5. Localizations of quotients of 𝑅_{𝑞}[𝐺] by its 𝐻-primes 1320
- 2.6. Spectral decomposition theorem for 𝑅_{𝑞}[𝐺] 1421
- 2.7. The De Concini–Kac–Procesi algebras 1623
- 2.8. A second presentation of 𝒰^{𝓌}_{±} 1724

- Chapter 3. A description of the centers of Joseph’s localizations 2128
- 3.1. Statement of the main result 2128
- 3.2. Associated root and weight spaces 2229
- 3.3. One side inclusion in \thref{BFW-CENTER} 2330
- 3.4. Joseph’s description of 𝑅_{𝐰} 2330
- 3.5. Homogeneous 𝑃-normal elements of the algebras 𝑆^{±}_{𝑤_{±}} 2633
- 3.6. Homogeneous 𝑃-normal elements of the algebras 𝑆_{𝐰} 2936
- 3.7. Proof of \thref{BFW-CENTER} 3239

- Chapter 4. Primitive ideals of 𝑅_{𝑞}[𝐺] and a Dixmier map for 𝑅_{𝑞}[𝐺] 3744
- 4.1. A formula for the primitive ideals of 𝑅_{𝑞}[𝐺] 3744
- 4.2. Structure of 𝑝𝑟𝑖𝑚_{𝐰}𝐑_{𝐪}[𝐆] as a 𝕋^{𝕣}×𝕋^{𝕣}-homogeneous space 3845
- 4.3. The standard Poisson Lie structure on 𝐺 and its symplectic leaves 3946
- 4.4. Equations for the symplectic leaves of (𝐺^{𝐰},𝜋_{𝐆}) 4047
- 4.5. A 𝕋^{𝕣}×𝕋^{𝕣}-equivariant Dixmier map for ℝ_{𝕢}[𝔾] 4148

- Chapter 5. Separation of variables for the algebras 𝑆^{±}_{𝑤} 4350
- Chapter 6. A classification of the normal and prime elements of the De Concini–Kac–Procesi algebras 4956
- 6.1. Statement of the classification result 4956
- 6.2. Homogeneous normal and 𝑃-normal elements of 𝑆^{±}_{𝑤} 5158
- 6.3. A lemma on diagonal automorphisms of 𝒰^{𝓌}_{±} 5259
- 6.4. Proof of \prref{NORMALP} 5461
- 6.5. Proof of \thref{NORMAL1} 5562
- 6.6. Prime and primitive ideals in the {0}-stratum of 𝑆𝑝𝑒𝑐𝑆^{±}_{𝑤}. 5663
- 6.7. A classification of the prime elements of 𝑆^{±}_{𝑤} 6269

- Chapter 7. Module structure of 𝑅_{𝐰} over their subalgebras generated by Joseph’s normal elements 6774
- Chapter 8. A classification of maximal ideals of 𝑅_{𝑞}[𝐺] and a question of Goodearl and Zhang 7582
- Chapter 9. Chain properties and homological applications 8592
- Bibliography 8996