Quantum groups and their primitive ideals.

*(English)*Zbl 0808.17004
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 29. Berlin: Springer-Verlag. ix, 383 p. (1995).

This is a text on the primitive ideal theory on the enveloping algebras \(U_ q ({\mathfrak g})\) attached to quantum groups \({\mathfrak G}\), focussing on the relationships between these and ordinary enveloping algebras \(U({\mathfrak g})\) (of semisimple Lie algebras \({\mathfrak g}\)) rather than on quantum groups themselves. The main objective is to give a detailed analysis of the structure of \(U_ q( {\mathfrak g})\) and a closely related Hopf algebra \(R_ q(G)\) and in particular to describe their primitive and prime spectra. Here \(R_ q(G)\) is a quantum analogue of the usual algebra \(R[G]\) of regular functions on an affine algebraic group \(G\). Many of the results stem from the author’s joint work with G. Letzter.

Chapter 1 gives an introduction to the theory of Hopf algebras emphasizing those results needed later on in the book. Chapter 2 redoes some of the classical theory of algebraic groups and their Lie algebras from a Hopf algebra point of view. Chapters 3 and 4 show how to generate quantum enveloping algebras and Kac-Moody Lie algebras from their common source, namely (generalized) Cartan matrices, and also develops some of the theory of integrable highest weight modules. Chapters 5 and 6 introduce some remarkable bases of \(U_ q ({\mathfrak g})\) which play a key role in the structure theory of the latter. These bases actually have classical analogues in \(U({\mathfrak g})\) whose existence had been previously unsuspected. Finally, Chapter 7 through 10 present the main results. Some miscellaneous algebraic and combinatorial facts needed at various places are summarized in appendices, together with historical sketch of the theory of quantum groups, which is otherwise rather neglected in the book.

The book is in the tradition of Dixmier’s and Jantzen’s earlier books on enveloping algebras. The author expresses the hope that quantum groups may shed some light on current unsolved problems in enveloping algebras.

Chapter 1 gives an introduction to the theory of Hopf algebras emphasizing those results needed later on in the book. Chapter 2 redoes some of the classical theory of algebraic groups and their Lie algebras from a Hopf algebra point of view. Chapters 3 and 4 show how to generate quantum enveloping algebras and Kac-Moody Lie algebras from their common source, namely (generalized) Cartan matrices, and also develops some of the theory of integrable highest weight modules. Chapters 5 and 6 introduce some remarkable bases of \(U_ q ({\mathfrak g})\) which play a key role in the structure theory of the latter. These bases actually have classical analogues in \(U({\mathfrak g})\) whose existence had been previously unsuspected. Finally, Chapter 7 through 10 present the main results. Some miscellaneous algebraic and combinatorial facts needed at various places are summarized in appendices, together with historical sketch of the theory of quantum groups, which is otherwise rather neglected in the book.

The book is in the tradition of Dixmier’s and Jantzen’s earlier books on enveloping algebras. The author expresses the hope that quantum groups may shed some light on current unsolved problems in enveloping algebras.

Reviewer: W.M.McGovern (Seattle)

##### MSC:

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |