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Extended Affine Lie Algebras and Their Root Systems
Bruce N. Allison, University of Alberta, Edmonton, AB, Canada, Saeid Azam and Stephen Berman, University of Saskatchewan, Saskatoon, SK, Canada, and Yun Gao and Arturo Pianzola, University of Alberta, Edmonton, AB, Canada

Memoirs of the American Mathematical Society
1997; 122 pp; softcover
Volume: 126
ISBN-10: 0-8218-0594-0
ISBN-13: 978-0-8218-0594-7
List Price: US$48
Individual Members: US$28.80
Institutional Members: US$38.40
Order Code: MEMO/126/603
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This work is about extended affine Lie algebras (EALA's) and their root systems. EALA's were introduced by Høegh-Krohn and Torresani under the name irreducible quasi-simple Lie algebras. The major objective is to develop enough theory to provide a firm foundation for further study of EALA's.

The first chapter of the paper is devoted to establishing some basic structure theory. It includes a proof of the fact that, as conjectured by Kac, the invariant symmetric bilinear form on an EALA can be scaled so that its restriction to the real span of the root system is positive semi-definite.

The second chapter studies extended affine root systems (EARS) which are an axiomatized version of the root systems arising from EALA's. The concept of a semilattice is used to give a complete description of EARS. In the final chapter, a number of new examples of extended affine Lie algebras are given.

The concluding appendix contains an axiomatic characterization of the nonisotropic roots in an EARS in a more general context than the one used in the rest of the paper.


  • Provides a foundation for the study of an important class of Lie algebras that generalizes the class of affine Kac-Moody Lie algebras
  • Includes material on Lie algebras and on root systems that can be read independently.


Graduate students, research mathematicians and mathematical physicists interested in Lie theory.

Table of Contents

  • Introduction
  • The basic structure theory of extended affine Lie algebras
  • Semilattices and extended affine root systems
  • Examples of extended affine Lie algebras
  • Appendix. Axiomatic theory of roots
  • References
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