Gebert, Reinhold W. Introduction to vertex algebras, Borcherds algebras and the monster Lie algebra. (English) Zbl 0860.17039 Int. J. Mod. Phys. A 8, No. 31, 5441-5503 (1993). Summary: The theory of vertex algebras constitutes a mathematically rigorous axiomatic formulation of the algebraic origins of conformal field theory. In this context Borcherds algebras arise as certain “physical” subspaces of vertex algebras. The aim of this review is to give a pedagogical introduction to this rapidly developing area of mathematics. Based on the machinery of formal calculus, we present the axiomatic definition of vertex algebras. We discuss the connection with conformal field theory by deriving important implications of these axioms. In particular, many explicit calculations are presented to stress the eminent role of the Jacobi identity axiom for vertex algebras. As a class of concrete examples the vertex algebras associated with even lattices are constructed and it is shown in detail how affine Lie algebras and the fake monster Lie algebra naturally appear. This leads us to the abstract definition of Borcherds algebras as generalized Kac-Moody algebras and their basic properties. Finally, the results about the simplest generic Borcherds algebras are analyzed from the point of view of symmetry in quantum theory and the construction of the monster Lie algebra is sketched. Cited in 1 ReviewCited in 10 Documents MSC: 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B69 Vertex operators; vertex operator algebras and related structures Keywords:Borcherds algebra; vertex algebra × Cite Format Result Cite Review PDF Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Expansion of 1/eta(q)^24; Fourier coefficients of T_{14}.