H_T Vertex Algebras and the Infinite Toda Lattice

2005/05/13 by Maarten Bergvelt, Bergvelt, Maarten
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #MSC-class: 17B69 #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #math-ph #math.MP #math.QA #msc:17B69

paper · pdf · doi:10.48550/arxiv.math/0505289

97 pages

arxiv created 2005/05/13 · arxiv updated 2009/12/01

Abstract

Let H_T=C[T,T^-1] be the Hopf algebra of symmetries of a lattice of rank 1, or equivalently, H_T is the group algebra of a free Abelian group with one generator T. We construct conformal algebras, vertex Poisson algebras and vertex algebras with H_T as symmetry. For example, the Hamiltonian structure for the infinite Toda lattice gives rise to an H_T-vertex Poisson structure on a free difference algebra. Examples of H_T-vertex algebras are constructed from representations of a class of infinite dimensional Lie algebras related to H_T in the same way loop algebras are related to the Hopf algebra H_D=C[D] of infinitesimal translations used in the usual vertex algebras.

Related

• H_T Vertex Algebras
• A unified construction of vertex algebras from infinite-dimensional Lie algebras
• Deformation quantization of vertex Poisson algebras
• The crystal duality principle: From Hopf algebras to geometrical symmetries
• ℏ-Vertex algebras and chiralization of star products