A new class of infinite-dimensional Lie algebras (continuum Lie algebras) and associated nonlinear systems

Abstract

We give a review of our recent results concerning a new class of infinite-dimensional Lie algebras — the generalizations of Z-graded contragredient Lie algebras with a, generally speaking, infinite-dimensional Cartan subalgebra and a contiguous set of roots (a manifold or a more general space, for example with a measure). We call such algebras “continuum Lie algebras”. Special examples of these algebras are the Kac-Moody algebras, the Poisson bracket algebras, algebras of vector fields on a manifold, current algebras, various versions of gl(∞), algebras of diffeomorphisms of a manifold, more general cross product Lie algebras (including, in particular, various multiindex generalizations of the Virasoro algebra), and other Lie algebras with differential or integro-differential Cartan operator (a continuous extension of the generalized Cartan matrix).