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From Dynkin diagram symmetries to fixed point structures

J"urgen Fuchs, Bert Schellekens, Christoph Schweigert

Abstract

Any automorphism of the Dynkin diagram of a symmetrizable Kac-Moody algebra induces an automorphism of the algebra and a mapping between its highest weight modules. For a large class of such Dynkin diagram automorphisms, we can describe various aspects of these maps in terms of another Kac-Moody algebra, the `orbit Lie algebra'. In particular, the generating function for the trace of the map on modules, the `twining character', is equal to a character of the orbit Lie algebra. Orbit Lie algebras and twining characters constitute a crucial step towards solving the fixed point resolution problem in conformal field theory.

From Dynkin diagram symmetries to fixed point structures

Abstract

Any automorphism of the Dynkin diagram of a symmetrizable Kac-Moody algebra induces an automorphism of the algebra and a mapping between its highest weight modules. For a large class of such Dynkin diagram automorphisms, we can describe various aspects of these maps in terms of another Kac-Moody algebra, the `orbit Lie algebra'. In particular, the generating function for the trace of the map on modules, the `twining character', is equal to a character of the orbit Lie algebra. Orbit Lie algebras and twining characters constitute a crucial step towards solving the fixed point resolution problem in conformal field theory.

Paper Structure

This paper contains 41 sections, 309 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Orbit Lie algebra s
  3. Twining character s
  4. Fixed point resolution
  5. Organization
  6. Folding Cartan matrices
  7. Dynkin diagram automorphisms
  8. The folded Cartan matrix
  9. Type conservation
  10. Simple Cartan matrices
  11. Affine Cartan matrices
  12. Hyperbolic Cartan matrices
  13. Lie algebra automorphisms
  14. Symmetrizable Kac$\hbox{- -}$ Moo-dy algebra s
  15. Induced outer automorphisms
  16. ...and 26 more sections

Figures (2)

  • Figure 2: The foldings of Dynkin diagram s with $\breve A^{i,j}_{}\breve A^{j,i}_{}\le3$.
  • Figure :