H_T Vertex Algebras and the Infinite Toda Lattice

Details H_T Vertex Algebras and the Infinite Toda Lattice H_T Vertex Algebras and the Infinite Toda Lattice

2005-05-13

arxiv.org/abs/math/0505289v1

Mathematics

Quantum Algebra

97 pages

Scientific paper

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.

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