© 2017

Representations of Lie Algebras and Partial Differential Equations


Table of contents

  1. Front Matter
    Pages i-xxxvi
  2. Fundament of Lie Algebras

    1. Front Matter
      Pages 1-1
    2. Xiaoping Xu
      Pages 3-32
    3. Xiaoping Xu
      Pages 33-59
    4. Xiaoping Xu
      Pages 61-93
    5. Xiaoping Xu
      Pages 125-151
  3. Explicit Representations

    1. Front Matter
      Pages 153-153
    2. Xiaoping Xu
      Pages 155-216
    3. Xiaoping Xu
      Pages 293-328
    4. Xiaoping Xu
      Pages 329-349
    5. Xiaoping Xu
      Pages 351-399
    6. Xiaoping Xu
      Pages 401-478
  4. Related Topics

    1. Front Matter
      Pages 479-479
    2. Xiaoping Xu
      Pages 523-554
    3. Xiaoping Xu
      Pages 555-616
  5. Back Matter
    Pages 617-620

About this book


This book provides explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic codes, combinatorics and algebraic varieties, summarizing the author’s works and his joint works with his former students.  Further, it presents various oscillator generalizations of the classical representation theorem on harmonic polynomials, and highlights new functors from the representation category of a simple Lie algebra to that of another simple Lie algebra.

Partial differential equations play a key role in solving certain representation problems. The weight matrices of the minimal and adjoint representations over the simple Lie algebras of types E and F are proved to generate ternary orthogonal linear codes with large minimal distances. New multi-variable hypergeometric functions related to the root systems of simple Lie algebras are introduced in connection with quantum many-body systems in one dimension. In addition, the book identifies certain equivalent combinatorial properties on representation formulas, and the irreducibility of representations is proved directly related to algebraic varieties. The book offers a valuable reference guide for mathematicians and scientists alike. As it is largely self-contained – readers need only a minimal background in calculus and linear algebra – it can also be used as a textbook.


Representations of Lie Algebras Special Linear Algebras Even Orthogonal Lie Algebras Odd Orthogonal Lie Algebras Symplectic Lie Algebras Codes Related to Representations Root-Related Integrable Systems

Authors and affiliations

  1. 1.Institute of MathematicsAcademy of Mathematics and Systems Science, Chinese Academy of SciencesBeijingChina

About the authors

In 1992, Xiaoping Xu obtained his Ph.D. from Rutgers University in United States. He had worked  at the Hong Kong University of Sciences and Technology from 1992 to 2002. He has been a professor at Institute of Mathematics of Chinese Academy of Sciences since 2002 and a professor at the University at Chinese Academy of Sciences since 2014.

Bibliographic information