# On the Spectra of Quantum Groups

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*Milen Yakimov*

#### 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