YANG-BAXTER AND KAC-MOODY ALGEBRAS IN FIELD THEORY WITH SOLITONS
Hector J. de Vega
The integrability of some important physical systems with a finite number of degrees of freedom relies on purely algebraic properties. One can solve the Coulomb potential and the harmonic oscillator both classically and quantum mechanically, using the appropriate dynamical algebra only [1]. Integrable systems with an infinite number of degrees of freedom are those with an infinite number of conserved quantities. They exhibit loop algebras, Yang-Baxter algebras and Kac-Moody algebras as symmetries. The investigation of theories has shown in recent years that Yang-Baxter algebras are the basic concepts of integrability [2]. The commutativity of a family of transfer matrices both in field theory and statistical mechanics follows directly from the Yang-Baxter algebra of monodromy matrices. Since the transfer matrix is a generating functional of local and non-local charges (including the hamiltonian) one gets an infinite set of commuting and conserved quantities. So, we can say that the theory is integrable. We are using the word integrable in the large sense explained above. Moreover, theories can be often explicitly solved to a large extent [2, 3]. In this brief report I will concentrate on a specific class of field theories. A complete account can be found in refs. 4-7. This class
Talk given at•Jan 1, 1985•0 citations