Quantum Lie theory : a multilinear approach / Vladislav Kharchenko.

Includes bibliographical references and index.

1. Elements of Noncommutative Algebra --
2. Poincare-Birkhoff-Witt Basis --
3. Quantizations of Kac-Moody Algebras --
4. Algebra of Skew-Primitive Elements --
5. Multilinear Operations --
6. Braided Hopf Algebras --
7. Binary Structures --
8. Algebra of Primitive Nonassociative Polynomials --
References --
Index.

This is an introduction to the mathematics behind the phrase “quantum Lie algebra”. The numerous attempts over the last 15-20 years to define a quantum Lie algebra as an elegant algebraic object with a binary “quantum” Lie bracket have not been widely accepted. In this book, an alternative approach is developed that includes multivariable operations. Among the problems discussed are the following: a PBW-type theorem; quantum deformations of Kac--Moody algebras; generic and symmetric quantum Lie operations; the Nichols algebras; the Gurevich--Manin Lie algebras; and Shestakov--Umirbaev operations for the Lie theory of nonassociative products. Opening with an introduction for beginners and continuing as a textbook for graduate students in physics and mathematics, the book can also be used as a reference by more advanced readers. With the exception of the introductory chapter, the content of this monograph has not previously appeared in book form.