Mathematical Institute of Russian Academy of Science,

112 Chernyshevski Str.,

450077 Ufa,

RUSSIA

E-mail: habibullin_i@mail.ru, ihabib@imat.rb.ru

**Characteristic algebras of the discrete equations**

**Abstract:**

It is well known that the characteristic Lie algebras, introduced by
A.B. Shabat in 1980, play the crucial role in studying the hyperbolic type
partial differential equations. For example, if the haracteristic algebra
of the equation is of finite dimension, then the equation is solved in
quadratures, if the algebra is of finite growth then the equation is integrated
by the inverse scattering method. Recently it has been observed by A.V.
Zhiber that the characteristic algebra provides an effective tool to classify
the nonlinear hyperbolic equations. However, the characteristic algebras
has not yet been used to study the discrete equations.

The notion of the characteristic Lie algebra of the discrete hyperbolic type equation is introduced. An effective algorithm to compute the algebra for the equation given is suggested. Examples and further applications are discussed.