On the Spectra of Quantum Groups.

Intro -- Contents -- Chapter 1. Introduction -- Chapter 2. Previous results on spectra of quantum function algebras -- 2.1. Quantized universal enveloping algebras -- 2.2. Type 1 modules and braid group action -- 2.3. -prime ideals of Quantum Groups -- 2.4. Sets of normal elements -- 2.5. Localizations of quotients of _{ }[ ] by its -primes -- 2.6. Spectral decomposition theorem for _{ }[ ] -- 2.7. The De Concini-Kac-Procesi algebras -- 2.8. A second presentation of ^{ }_{±} -- Chapter 3. A description of the centers of Joseph's localizations -- 3.1. Statement of the main result -- 3.2. Associated root and weight spaces -- 3.3. One side inclusion in \thref{BFW-CENTER} -- 3.4. Joseph's description of _{ } -- 3.5. Homogeneous -normal elements of the algebras ^{±}_{ _{±}} -- 3.6. Homogeneous -normal elements of the algebras _{ } -- 3.7. Proof of \thref{BFW-CENTER} -- Chapter 4. Primitive ideals of _{ }[ ] and a Dixmier map for _{ }[ ] -- 4.1. A formula for the primitive ideals of _{ }[ ] -- 4.2. Structure of _{ } _{ }[ ] as a ^{ }× ^{ }-homogeneous space -- 4.3. The standard Poisson Lie structure on and its symplectic leaves -- 4.4. Equations for the symplectic leaves of ( ^{ }, _{ }) -- 4.5. A ^{ }× ^{ }-equivariant Dixmier map for ℝ_{ }[ ] -- Chapter 5. Separation of variables for the algebras ^{±}_{ } -- 5.1. Statement of the freeness result -- 5.2. Leading terms of the normal elements _{ }^{±}( ^{±}_{ , }) -- 5.3. Proof of \thref{FREES} -- Chapter 6. A classification of the normal and prime elements of the De Concini-Kac-Procesi algebras -- 6.1. Statement of the classification result -- 6.2. Homogeneous normal and -normal elements of ^{±}_{ } -- 6.3. A lemma on diagonal automorphisms of ^{ }_{±} -- 6.4. Proof of \prref{NORMALP}.

6.5. Proof of \thref{NORMAL1} -- 6.6. Prime and primitive ideals in the {0}-stratum of ^{±}_{ }. -- 6.7. A classification of the prime elements of ^{±}_{ } -- Chapter 7. Module structure of _{ } over their subalgebras generated by Joseph's normal elements -- 7.1. Statement of the freeness result -- 7.2. A × -filtration of _{ } -- 7.3. The action of '_{ } on _{ } -- 7.4. Structure of the algebras '_{ } and separation of variables for _{ } -- 7.5. Structure of the algebras _{ } and freeness of _{ } over _{ } -- Chapter 8. A classification of maximal ideals of _{ }[ ] and a question of Goodearl and Zhang -- 8.1. A projection property of the ideal _{(1,1)} -- 8.2. Proof of \prref{PROJJ} -- 8.3. Proof of \thref{MAX} -- 8.4. Classification of _{ }[ ] and a question of Goodearl and Zhang -- Chapter 9. Chain properties and homological applications -- 9.1. Applications -- 9.2. ⁺⊛ ⁻ is an algebra with enough normal elements. -- Bibliography.

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