ABSTRACT
Comparing such relations with the commutation relations of the w-twisted loop algebra ~' one then has [39,44] Theorem 9. The operators {a(k), k E Z, a E Q0 } (creation-annihilation operators), {E~, i E Z, a E Q0 }, together with the identity operator, span a Lie algebra of operators acting on the Hilbert space '7JC ® S('d' _), is isomorphic with the w-twisted loop algebra
5.4 BEYOND AFFINE LIE ALBEGRAS Affine Lie algebras have been shmvn to possess a very rich (highest weight) representation theory, and as a consequence, a very large domain of applications in mathematics and physics. Let us quote for instance the do-
(i) C§ is provided with a nondegenerate invariant symmeric bilinear form, called the Killing form, and denoted by ( ... ). For simplicity, the Killing form will be assumed to be real on the real linear span of the root system.
