nLab affine Lie algebra



Lie theory

∞-Lie theory (higher geometry)


Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids



Related topics


\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras




Affine Lie algebras (sometimes: current 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.


Textbook account

Lecture notes:

The standard textbook on loop groups is

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

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

  • 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)

and specifically a review in the context of the Witten genus is in

  • Kefeng Liu, section 2.2 of On modular invariance and rigidity theorems, J. Differential Geom. Volume 41, Number 2 (1995), 247-514 (EUCLID, pdf)

The famous quote by Kac is from:

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.

Relation to modular forms:


On non-integrable but “admissible” irreps of affine Lie algebras:

For the special case 𝔤=\mathfrak{g} = 𝔰𝔩 ( 2 , ) \mathfrak{sl}(2, \mathbb{C}) the formula for the “admissible” weights is made explicit in

  • Boris Feigin, Fyodor Malikov, Modular functor and representation theory of 𝔰𝔩 2^\widehat{\mathfrak{sl}_2} at a rational level, p. 357-405 in: Loday, Stasheff, Voronov (eds.) Operads: Proceedings of Renaissance Conferences, Contemporary Mathematics 202 , AMS (1997) [arXiv:q-alg/9511011, ams:conm-202]

Last revised on January 19, 2023 at 17:31:41. See the history of this page for a list of all contributions to it.