Integrable representations and the Weyl group of a Kac-Moody algebra
In this chapter we begin a systematic study of the Kac-Moody algebras. Recall that this is the Lie algebra g(A) associated to a generalized Cartan matrix A. The main object of the chapter is the Weyl group W of a Kac-Moody algebra, which is a generalization of the classical Weyl group in the finite-dimensional theory. However, in contrast to the finite-dimensional case, W is infinite and the union of the W-translates of the fundamental chamber is a convex cone, not the whole Cartan subalgebra h.
Unable to display preview. Download preview PDF.
Bibliographical notes and comments
- Kac, V. G. [1968 B] Simple irreducible graded Lie algebras of finite growth, English translation: Math. USSRIzvestija 2 (1968), 1271–1311.Google Scholar
- Tits, J.  Resumé de cours, Annuaire du Collège de France 1980–81, Collège de France, Paris.Google Scholar
- Vinberg, E. B.  Discrete linear groups generated by reflections, English translation: Math. USSRIzvestija 5 (1971), 1083–1119.Google Scholar
- Kac, V. G., Peterson, D. H. [1983 A] Infinite dimensional Lie algebras, theta functions and modular forms, Advances in Math., 50 (1983).Google Scholar
- Piatetsky-Shapiro, I. I., Shafarevich, I. R.  A Torelli theorem for algebraic surfaces of type K3, Izvestija AN USSR (Ser. Mat.) 35 (1971), 530–572.Google Scholar
- Kac, V. G., Peterson, D. H. [1983 B] Regular functions on certain infinite dimensional groups, in Arithmetic and Geometry, (ed. M. Artin and J. Tate ), 141–166, Birkhäuser, Boston, 1983.Google Scholar
- Kac, V. G., Peterson, D. H. [1983 C] Unitary structure in representations of infinite-dimensional groups and a convexity theorem, MIT, preprint.Google Scholar
- Kac, V. G. [1969 B] An algebraic definition of compact Lie groups, Trudy MIEM, No. 5, 1969, 36–47 (in Russian).Google Scholar