Infinite Dimensional Lie Algebras

TLDR: The invariant bilinear form and the generalized casimir operator are integral representations of Kac-Moody algebras and the weyl group as mentioned in this paper, as well as a classification of generalized cartan matrices.

Abstract: Introduction Notational conventions 1 Basic definitions 2 The invariant bilinear form and the generalized casimir operator 3 Integrable representations of Kac-Moody algebras and the weyl group 4 A classification of generalized cartan matrices 5 Real and imaginary roots 6 Affine algebras: the normalized cartan invariant form, the root system, and the weyl group 7 Affine algebras as central extensions of loop algebras 8 Twisted affine algebras and finite order automorphisms 9 Highest-weight modules over Kac-Moody algebras 10 Integrable highest-weight modules: the character formula 11 Integrable highest-weight modules: the weight system and the unitarizability 12 Integrable highest-weight modules over affine algebras 13 Affine algebras, theta functions, and modular forms 14 The principal and homogeneous vertex operator constructions of the basic representation Index of notations and definitions References Conference proceedings and collections of paper