© 2015

Quantum Lie Theory

A Multilinear Approach


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

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Vladislav Kharchenko
    Pages 1-69
  3. Vladislav Kharchenko
    Pages 71-97
  4. Vladislav Kharchenko
    Pages 99-127
  5. Vladislav Kharchenko
    Pages 129-150
  6. Vladislav Kharchenko
    Pages 151-198
  7. Vladislav Kharchenko
    Pages 199-244
  8. Vladislav Kharchenko
    Pages 245-273
  9. Vladislav Kharchenko
    Pages 275-287
  10. Back Matter
    Pages 289-304

About this book


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

Authors and affiliations

  1. 1.Universidad Nacional Autónoma de MéxicoCuautitlán IzcalliMexico

Bibliographic information


“The contents of the chapters 2-8 are mostly based on papers of the author and are here presented as a unified, self-contained whole with the first chapter containing the necessary preliminaries, making the exposition more accessible to a broader audience. … Each chapter ends with detailed notes providing pointers to literature and some open questions related to the presented material are discussed as well.” (Isar Rober Goyvaerts, zbMATH 1337.17001, 2016)