The Variety of Leibniz Algebras Defined by the Identity x(y(zt)) ≡ 0

2000 Mathematics Subject Classification: Primary: 17A32; Secondary: 16R10, 16P99, 17B01, 17B30, 20C30

Let F be a field of characteristic zero. In this paper we study the variety of Leibniz algebras 3N determined by the identity x(y(zt)) ≡ 0. The algebras of this variety are left nilpotent of class not more than 3. We give a complete description of the vector space of multilinear identities in the language of representation theory of the symmetric group Sn and Young diagrams. We also show that the variety 3N is generated by an abelian extension of the Heisenberg Lie algebra. It has turned out that 3N has many properties which are similar to the properties of the variety of the abelian-by-nilpotent of class 2 Lie algebras. It has overexponential growth of the codimension sequence and subexponential growth of the colength sequence.

This project was partially supported by RFBR, grants 01-01-00728, 02-01-00219 and UR 04.01.036.