School of Mathematics,

University of Leeds,

Leeds LS2 9JT,

UK

E-mail: sashamik@maths.leeds.ac.uk

**Infinite Dimensional Automorphic Lie Algebras**

**Abstract:**

We propose a new class of infinite dimensional Lie algebras, which
has important applications to the theory of integrable equations. The construction
of these algebras is very similar to the one for automorphic functions
and this motivates the name automorphic Lie algebras. For automorphic Lie
algebras we present bases in which they are quasigraded and all structure
constants can be written out explicitly. These algebras have a useful factorisations
on two subalgebras similar to the factorisation of the current algebra
on the positive and negative parts.