H_T Vertex Algebras

Abstract

The usual vertex algebras have as underlying symmetry the Hopf algebra $H_D=\mathbb{C}[D]$ of infinitesimal translations. We show that it is possible to replace $H_D$ by another symmetry algebra $H_T=\mathbb C[T,T\inv]$, the group algebra of the Abelian group generated by $T$. $H_T$ is the algebra of symmetries of a lattice of rank 1, and the construction gives a class of vertex algebras related to the Infinite Toda Lattice in the same way as the usual $H_D$-vertex algebras are related to Korteweg-de Vries hierarchies.

Contribution to the Proceedings of the Conference "Lie Algebras, Vertex Operator Algebras and Their Applications" in honor of Jim Lepowsky and Robert Wilson