Author	Joseph, Anthony. author
Title	Quantum Groups and Their Primitive Ideals [electronic resource] / by Anthony Joseph
Imprint	Berlin, Heidelberg : Springer Berlin Heidelberg, 1995
Connect to	http://dx.doi.org/10.1007/978-3-642-78400-2
Descript	IX, 383 p. online resource

by a more general quadratic algebra (possibly obtained by deformation) and then to derive Rq [G] by requiring it to possess the latter as a comodule. A third principle is to focus attention on the tensor structure of the catยญ egory of (!; modules. This means of course just defining an algebra structure on Rq[G]; but this is to be done in a very specific manner. Concretely the category is required to be braided and this forces (9.4.2) the existence of an "R-matrix" satisfying in particular the quantum Yang-Baxter equation and from which the algebra structure of Rq[G] can be written down (9.4.5). Finally there was a search for a perfectly self-dual model for Rq[G] which would then be isomorphic to Uq(g). Apparently this failed; but V. G. Drinfeld found that it could be essentially made to work for the "Borel part" of Uq(g) denoted U (b) and further found a general construction (the Drinfeld double) q mirroring a Lie bialgebra. This gives Uq(g) up to passage to a quotient. One of the most remarkable aspects of the above superficially different apยญ proaches is their extraordinary intercoherence. In particular they essentially all lead for G semisimple to the same and hence "canonical", objects Rq[G] and Uq(g), though this epithet may as yet be premature

I. Hopf Algebras -- 1.1 Axioms of a Hopf Algebra -- 1.2 Group Algebras and Enveloping Algebras -- 1.3 Adjoint Action -- 1.4 The Hopf Dual -- 1.5 Comments and Complements -- 2. Excerpts from the Classical Theory -- 2.1 Lie Algebras -- 2.2 Algebraic Lie Algebras -- 2.3 Algebraic Groups -- 2.4 Lie Algebras of Algebraic Groups -- 2.5 Comments and Complements -- 3. Encoding the Cartan Matrix -- 3.1 Quantum Weyl Algebras -- 3.2 The Drinfeld Double -- 3.3 The Rosso Form and the Casimir Invariant -- 3.4 The Classical Limit and the Shapovalev Form -- 3.5 Comments and Complements -- 4. Highest Weight Modules -- 4.1 The Jantzen Filtration and Sum Formula -- 4.2 Kac-Moody Lie Algebras -- 4.3 Integrable Modules for Uq(gc) -- 4.4 Demazure Modules and Product Formulae -- 4.5 Comments and Complements -- 5. The Crystal Basis -- 5.1 Operators in the Crystal Limit -- 5.2 Crystals -- 5.3 Ad-invariant Filtrations, Twisted Actions and the Crystal Basis for Uq(n-) -- 5.4 The Grand Loop -- 5.5 Comments and Complements -- 6. The Global Bases -- 6.1 The ? Operation and the Embedding Theorem -- 6.2 Globalization -- 6.3 The Demazure Property -- 6.4 Littelmannโ{128}{153}s Path Crystals -- 6.5 Comments and Complements -- 7. Structure Theorems for Uq(g) -- 7.1 Local Finiteness for the Adjoint Action -- 7.2 Positivity of the Rosso Form -- 7.3 The Separation Theorem -- 7.4 Noetherianity -- 7.5 Comments and Complements -- 8. The Primitive Spectrum of Uq(g) -- 8.1 The Poincarรฉ Series of the Harmonic Space -- 8.2 Factorization of the Quantum PRV Determinants -- 8.3 Verma Module Annihilators -- 8.4 Equivalence of Categories -- 8.5 Comments and Complements -- 9. Structure Theorems for Rq[G] -- 9.1 Commutativity Relations -- 9.2 Surjectivity and Injectivity Theorems -- 9.3 The Adjoint Action -- 9.4 The R-Matrix -- 9.5 Comments and Complements -- 10. The Prime Spectrum of Rq[G] -- 10.1 Highest Weight Modules -- 10.2 The Quantum Weyl Group -- 10.3 Prime and Primitive Ideals of Rq[G] -- 10.4 Hopf Algebra Automorphisms -- 10.5 Comments and Complements -- A.2 Excerpts from Ring Theory -- A.3 Combinatorial Identities and Dimension Theory -- A.4 Remarks on Constructions of Quantum Groups -- A.5 Comments and Complements -- Index of Notation