On the Connection of Certain Lie Algebras with Vertex Algebras

In this paper we survey the connection of certain infinite-dimensional Lie algebras, including twisted and untwisted affine Lie algebras, toroidal Lie algebras and quantum torus Lie algebras, with vertex algebras. Introduction Vertex (operator) algebras are a new class of algebraic structures and they have deep connections with numerous fields. In mathematics, vertex algebras have been a vibrant research area. On the other hand, as the algebraic counterpart of chiral algebras, vertex operator algebras together with their representations provide a solid foundation for the study of conformal field theory in physics. Though vertex algebras are highly non-classical, they have connections with classical algebras such as Lie algebras, associative algebras and groups. In particular, vertex algebras are often constructed and studied by using classical (infinite-dimensional) Lie algebras. For example, those vertex operator algebras associated to (untwisted) affine Kac-Moody Lie algebras (including infinite-dimension Heisenberg Lie algebras) and the Virasoro Lie algebra (cf. [FZ], [DL], [Li1], [LL]) are among the important examples. These two families of vertex operator algebras underline the algebraic study of the physical Wess-Zumino-Novikov-Witten model and the minimal models in conformal field theory, respectively. On the other hand, twisted affine Lie algebras (see [K1]) can be also associated with vertex operator algebras in terms of twisted modules (see [FLM], [Li2]). In the theory of Lie algebras, by generalizing the loop-realization of untwisted affine Lie algebras, one has toroidal Lie algebras, which are perfect central extensions of multi-loop Lie algebras.