nLab affine Lie algebra

∞-Lie theory

Examples

$\infty$-Lie algebras

Algebra

higher algebra

universal algebra

Contents

Idea

Affine Lie algebras are the most important class of Kac-Moody Lie algebras. They should be viewed as tangent Lie algebras to the loop groups, with a correction term which is sometimes related to quantization/quantum anomaly.

These affine Lie algebras appear in quantum field theory as the current algebras in the WZW model as well as in its “chiral halfs”, as such for instance in the heterotic string 2d CFT.

References

• David Hernandez, An introduction to affine Kac-Moody algebras, pdf
• Victor G. Kac, The idea of locality, q-alg/9709008

It is a well kept secret that the theory of Kac-Moody algebras has been a disaster. True, it is a generalization of a very important object, the simple finite-dimensional Lie algebras, but a generalization too straightforward to expect anything interesting from it. True, it is remarkable how far one can go with all these ei’s, fi’s and hi’s. Practically all, even most difficult results of finite-dimensional theory, such as the theory of characters, Schubert calculus and cohomology theory, have been extended to the general set-up of Kac-Moody algebras. But the answer to the most important question is missing: what are these algebras good for? Even the most sophisticated results, like the connections to the theory of quivers, seem to be just scratching the surface.

However, there are two notable exceptions. The best known one is, of course, the theory of affine Kac-Moody algebras. This part of the Kac-Moody theory has deeply penetrated many branches of mathematics and physics. The most important single reason for this success is undoubtedly the isomorphism of affine algebras and central extensions of loop algebras, often called current algebras. The second notable exception is provided by Borcherds’ algebras which are roughly speaking the spaces of physical states of certain chiral algebras.

• A. Pressley, Graeme Segal, Loop groups, Oxford Univ. Press 1988

The relation to quantum physics (WZW model) is highlighted in the texts

• S. Kass, R. V. Moody, J. Patera, Affine Lie Algebras, Weight Multiplicities, and Branching Rules

• Louise Dolan, The Beacon of Kac-Moody symmetry for physics, Notices of the AMS 1995 (pdf)

Revised on September 12, 2013 01:18:27 by Urs Schreiber (145.116.131.249)