Ebook: Quantum Groups and Their Primitive Ideals

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 cat­ 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 ap­ 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.

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The primary aim of this book is an in-depth study of the Drinfeld-Jimbo quantization Uq(g) of the enveloping algebra U(g) of a semisimple Lie algebra g and of the Hopf dual Rq (G) of Uq(g). The focus is on determining the primitive spectra of these rings. A systematic use of Hopf algebra structure, and in particular of adjoint action, is to be made. There will be an emphasis on "quantum phenomena" which are new features of Uq(g) and on how these can be used to even simplify the study of U(g). The reader will learn how the quantum viewpoint has revitalized the study of enveloping algebras and will become acquainted with proofs which have been developed over the last 20 years into a particularly efficient form. Many of the results are now only just being published in research journals.

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The primary aim of this book is an in-depth study of the Drinfeld-Jimbo quantization Uq(g) of the enveloping algebra U(g) of a semisimple Lie algebra g and of the Hopf dual Rq (G) of Uq(g). The focus is on determining the primitive spectra of these rings. A systematic use of Hopf algebra structure, and in particular of adjoint action, is to be made. There will be an emphasis on "quantum phenomena" which are new features of Uq(g) and on how these can be used to even simplify the study of U(g). The reader will learn how the quantum viewpoint has revitalized the study of enveloping algebras and will become acquainted with proofs which have been developed over the last 20 years into a particularly efficient form. Many of the results are now only just being published in research journals.
Content:
Front Matter....Pages I-IX
Introduction....Pages 1-7
Hopf Algebras....Pages 8-35
Excerpts from the Classical Theory....Pages 36-61
Encoding the Cartan Matrix....Pages 62-95
Highest Weight Modules....Pages 96-130
The Crystal Basis....Pages 131-160
The Global Bases....Pages 161-200
Structure Theorems for U q (g)....Pages 201-232
The Primitive Spectrum of U q (g)....Pages 233-261
Structure Theorems for R q [G]....Pages 262-293
The Prime Spectrum of R q[G]....Pages 294-325
Back Matter....Pages 326-383

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The primary aim of this book is an in-depth study of the Drinfeld-Jimbo quantization Uq(g) of the enveloping algebra U(g) of a semisimple Lie algebra g and of the Hopf dual Rq (G) of Uq(g). The focus is on determining the primitive spectra of these rings. A systematic use of Hopf algebra structure, and in particular of adjoint action, is to be made. There will be an emphasis on "quantum phenomena" which are new features of Uq(g) and on how these can be used to even simplify the study of U(g). The reader will learn how the quantum viewpoint has revitalized the study of enveloping algebras and will become acquainted with proofs which have been developed over the last 20 years into a particularly efficient form. Many of the results are now only just being published in research journals.
Content:
Front Matter....Pages I-IX
Introduction....Pages 1-7
Hopf Algebras....Pages 8-35
Excerpts from the Classical Theory....Pages 36-61
Encoding the Cartan Matrix....Pages 62-95
Highest Weight Modules....Pages 96-130
The Crystal Basis....Pages 131-160
The Global Bases....Pages 161-200
Structure Theorems for U q (g)....Pages 201-232
The Primitive Spectrum of U q (g)....Pages 233-261
Structure Theorems for R q [G]....Pages 262-293
The Prime Spectrum of R q[G]....Pages 294-325
Back Matter....Pages 326-383
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