# Quantum Lie Theory

## A Multilinear Approach

Part of the Lecture Notes in Mathematics book series (LNM, volume 2150)

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Part of the Lecture Notes in Mathematics book series (LNM, volume 2150)

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.

17B37,20G42,16T20,16T05,17A50,17B75,17B81,17B81,81R50 Nichols algebra Poincaré-Birkhoff-Witt basis, quantum Lie operation, braided space character Hopf algebra

- DOI https://doi.org/10.1007/978-3-319-22704-7
- Copyright Information Springer International Publishing Switzerland 2015
- Publisher Name Springer, Cham
- eBook Packages Mathematics and Statistics
- Print ISBN 978-3-319-22703-0
- Online ISBN 978-3-319-22704-7
- Series Print ISSN 0075-8434
- Series Online ISSN 1617-9692
- Buy this book on publisher's site