nLab affine Lie algebra

Idea

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.

Related concepts

• Kac-Weyl character

• quantum affine algebra

• Borcherds algebra

• double affine Hecke algebra

• su(2)-anyon

References

Textbook account

• Victor G. Kac, Infinite Dimensional Lie Algebras, Progress in Mathematics 44 Springer 1983 [doi:10.1007/978-1-4757-1382-4], Cambridge University Press (1990) [doi:10.1017/CBO9780511626234]

Lecture notes:

• Minoru Wakimoto, Lectures on Infinite-Dimensional Lie Algebra, World Scientific (2001) [doi:10.1142/4269]

• David Hernandez, An introduction to affine Kac-Moody algebras (2006) (pdf)

• Iain Gordon, Infinite-dimensional Lie algebras (2008/9) (pdf

• Antony Wassermann, Kac-Moody and Virasoro algebras, course notes (2011) (pdf)

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:

• Victor 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.

Relation to modular forms:

• Victor G. Kac, Dale H. Peterson, Affine Lie algebras and Hecke modular forms, Bull. Amer. Math. Soc. (N.S.) 3 3 (1980) 1057-1061 [bams:1183547694]

• Victor G. Kac, Dale H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Advances in Mathematics 53 2 (1984) 125-264 [doi:10.1016/0001-8708(84)90032-X]

review:

• Ian G. MacDonald, Affine Lie algebras and modular forms, Séminaire Bourbaki: vol. 1980/81, exposés 561-578, Séminaire Bourbaki, no. 23 (1981), Exposé no. 577 [numdam:SB_1980-1981__23__258_0]

• Victor G. Kac, Minoru Wakimoto, Modular and conformal invariance constraints in representation theory of affine algebras, Advances in Mathematics 70 2 (1988) 156-236 [doi:10.1016/0001-8708(88)90055-2, spire:275458]

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

• Victor G. Kac, Minoru Wakimoto, Modular invariant representations of infinite-dimensional Lie algebras and superalgebras, PNAS 85 14 (1988) 4956-4960 [doi:10.1073/pnas.85.14.4956]

• Victor G. Kac, Minoru Wakimoto, Classification of modular invariant representations of affine algebras, p. 138-177 in V. G. Kač (ed.): Infinite dimensional Lie algebras and groups, Advanced series in Mathematical physics 7, World Scientific (1989) [pdf, cds:268092]

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

• Boris Feigin, Fyodor Malikov, Modular functor and representation theory of $\widehat{\mathfrak{sl}_2}$ 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]