Quantum Lie Theory: A Multilinear Approach

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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.