Construction of Finitely Presented Lie Algebras and Superalgebras

Abstract

We consider the following problem: what is the most general Lie algebra 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 algorithmic 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 integrability analysis of nonlinear partial differential equations and investigation of Lie (super)algebras arising in different (super)symmetrical physical models. The finite presentations also indicate a way to q-quantize 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 reason one needs, in practice, to use a computer algebra tool. We describe here an algorithm for constructing the basis of a finitely presented Lie (super)algebra and its commutator table, and its implementation in C. Some computer results illustrating our algorithm and its actual implementation are also presented.

Minimal involutive bases

“…By transforming a given system into this form one can determine the dimension of the solution space and a set of initial conditions providing the existence of a uniquely defined and locally holomorphic solution [17,26,27,28]. Involutive algorithmic ideas may be also rather fruitful in constructing the canonical bases for finitely generated ideals in free Lie algebras and superalgebras [29].…”

Matemati̇k Eği̇ti̇mi̇ Programlarina Çok Boyutlu Bi̇r Yaklaşim: "Lie Cebi̇ri̇" Örneği̇

“…given by Gerdt and Kornyak (1996). Gerdt also gives a program written in C-language calculating the base of L .…”

A Presentation of the Free Lie Algebra M_2,m

“…Finding a finite presentation defining a given Lie algebra and the minimalization of this presentation are fundamental problems of Lie algebra presentations. Research studies related of this aspect include the works of R. Bryant (Bryant, 1999) [1] and V. P. Gerdt and V. Kornyak (Gerdt and Kornyak, 1996) [2]. Group case of these problems has been dealt in a number of papers (Gupta and Levin, 1986;Searby and Wamsley, 1972;Moghaddam, 1999;Wamsley, 1973) [3], [5], [6], [7].…”