Abstract
Let M2,m,3 be a free solvable nilpotent Lie algebra of rank 2 and nilpotency class m - 1. We show that M2,m,3 admits a minimal presentation whose set of defining relators consists of certain types of basic commutators using techniques in Grobner-Shirshov basis theory.
Highlights
Grobner-Shirsov basis of the free Lie algebra γm(F ) + δ3(F ) are found and are construct a presentation for the Lie algebra M2,m,3 defined by generators and defining relations, where F is a free Lie algebra of rank 2 over a field of characteristic zero
The technique of Grobner-Shirsov bases is very useful in the study of presentations of Lie algebras, associative algebras, groups, etc., by generators and defining relations
We say that A is presented by the generating set X and the set of defining relations R and use the notation A = K X | R for the presentation of A or, allowing some freedom in the notation A = K X | R = 0
Summary
The set G ⊂ U is called a Grobner basis of U (or a complete system of defining relations of the algebra A = K(X)/U ) if the sets of normal words with respect to G and U coincide. Consider the Lie algebra M2,m,3 (m > 8) defined by the presentation, M2,m,3 = x, y | γm(F ) + δ3(F ). 2.8 Lemma (The Composition Lemma) Let L(X) be a free Lie algebra and I be its ideal generated by a complete set S. Let us define free generating sets and Hall basis for γm(F ) which will used in this paper. We construct a Hall basis HCm on Cm for the Lie algebra γm(F ), by forming products of elements of Cm such that Cm is a set of free generators for γm(F ). We are going to investigate a presentation of M2,m,3 for m > 8
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