CONSTRUCTION OF FINITELY PRESENTED LIE-ALGEBRAS AND SUPERALGEBRAS

We consider the following problem: what is the most general Lie algebr a or superalgebra satisfying a given set of Lie polynomial equations? The presentation of Lie (super)algebras by a finite set of generators and defining relations is one of the most general mathematical and alg orithmic schemes of their analysis. That problem is of great practical importance covering applications ranging from mathematical physics to combinatorial algebra. Some particular applications are construction of prolongation algebras in the Wahlquist-Estabrook method for integra bility analysis of nonlinear partial differential equations and invest igation of Lie (super)algebras arising in different (super)symmetrical physical models. The finite presentations also indicate a way to q-qu antize Lie (super)algebras. To solve this problem, one should perform a large volume of algebraic transformations which is sharply increased with growth of the number of generators and relations. For this reaso n one needs, in practice, to use a computer algebra tool. We describe here an algorithm for constructing the basis of a finitely presented L ie (super)algebra and its commutator table, and its implementation in C. Some computer results illustrating our algorithm and its actual imp lementation are also presented. (C) 1996 Academic Press Limited