Hierarchy structure in integrable systems of gauge fields and underlying Lie algebras

Abstract

An improved version of Nakamura's self-dual Yang-Mills hierarchy is presentd and its symmetry contents are studied. The new hierarchy as well as the previous one represents a set of commuting dynamical flows in an infinite dimensional manifolds of “loop type”, but includes a large set of dependent variables. Because of new degrees of freedom the theory acquires a more symmetric form with richer structures. For example it allows a large symmetry algebra of Riemann-Hilbert type, which is actually a direct sum of two subalgebras (“left” and “right”). This phenomenon is basically the same as observed recently by Avan and Bellon on the case of principal chiral models. In addition to these rather familiar symmeties, a new type of symmetries referred to as “coordinate transformation type” are also introduced. Generators of the above dynamical flows are all included therein. These two types of symmetries altogether form a big Lie algebra, which lead to more satisfactory understanding of symmetry properties of integrable systems of guage fields.