Affine Lie algebras: the normalized invariant bilinear form, the root system and the Weyl group
The results of Chapter 4 show that a Kac-Moody algebra g(A) is finite-dimensional if and only if all principal minors of A are positive. These Lie algebras are semisimple; moreover, by the classical structure theory, they exhaust all finite-dimensional semisimple Lie algebras. So, the classical Killing-Cartan theory of simple Lie algebras is, in our terminology, the theory of Kac-Moody algebras associated to a matrix of finite type.
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Bibliographical notes and comments
- Kac, V. G. [1968B] Simple irreducible graded Lie algebras of finite growth, Izvestija AN USSR (ser. mat.) 32 (1968), 1923-1967. English translation: Math. USSRIzvestija 2 (1968), 1271 - 1311.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