, denoted where is the complex representation with basis and define the action of on by (with ) Affine Vertex Algebra The vacuum representation in fact can be equipped with vertex algebra structure, in which case it is called the affine vertex algebra of rank . The affine Lie algebra naturally extends to the Kac–Moody algebra, with the differential represented by the translation operator in the vertex algebra. Weyl group and characters The Weyl group of an affine Lie algebra can be written as a semi-direct product of the Weyl group of the zero-mode algebra (the Lie algebra used to define the loop algebra) and the coroot lattice. The Weyl character formula of the algebraic characters of the affine Lie algebras generalizes to the Weyl-Kac character formula. A number of interesting constructions follow from these. One may construct generalizations of the Jacobi theta function. These theta functions transform under the modular group. The usual denominator identities of semi-simple Lie algebras generalize as well; because the characters can be written as "deformations" or q-analogs of the highest weights, this led to many new combinatoric identities, include many previously unknown identities for the Dedekind eta function. These generalizations can be viewed as a practical example of the Langlands program. Applications Due to the Sugawara construction, the universal enveloping algebra of any affine Lie algebra has the Virasoro algebra as a subalgebra. This allows affine Lie algebras to serve as symmetry algebras of conformal field theories such as WZW models or coset models. As a consequence, affine Lie algebras also appear in the worldsheet description of string theory. Example The Heisenberg algebra defined by generators satisfying commutation relations can be realized as the affine Lie algebra . References Lie algebras Representation theory