Lie algebras, variety of

over a ring

A class of Lie algebras (cf. Lie algebra) over that satisfy a fixed system of identities. The most prevalent varieties of Lie algebras are the following: the variety of Abelian Lie algebras specified by the identity , the variety of nilpotent Lie algebras of class (in which any products of length greater than are equal to zero), the variety of solvable Lie algebras of length (in which the derived series converges to zero in no more than steps). The totality of all varieties of Lie algebras over is a groupoid with respect to multiplication: , where is the class of extensions of algebras from by means of ideals from ; ; the algebras of are called metabelian.

The central problem in the theory of varieties of Lie algebras is to describe bases of identities of a variety of Lie algebras, in particular whether they are finite or infinite (if is a Noetherian ring). If is a field of characteristic , there are examples of locally finite varieties of Lie algebras lying in and not having a finite basis of identities. In the case of a field of characteristic 0 there are no examples up till now (1989) of infinitely based varieties. The finite basis property is preserved under right multiplication by a nilpotent variety and under union with such a variety. Among the Specht varieties (that is, those in which every variety is finitely based) are the varieties of Lie algebras over any Noetherian ring, over any field of characteristic , and , defined by identities that are true in the Lie algebra of matrices of order 2 over a field with . Over a field of characteristic 0 there are still no examples of a finite-dimensional Lie algebra such that is infinitely based, but there are such examples over an infinite field of characteristic . Over a finite field, or, more generally, over any finite ring with a unit, the identities of a finite Lie algebra follow from a finite subsystem of them.

A variety of Lie algebras generated by a finite algebra is called a Cross variety and is contained in a Cross variety consisting of Lie algebras in which all principal factors have order , all nilpotent factors have class and all inner derivations are annihilated by a unitary polynomial . Just non-Cross varieties (that is, non-Cross varieties all proper subvarieties of which are Cross varieties) have been described in the solvable case, and there are examples of non-solvable just non-Cross varieties. The groupoid over an infinite field is a free semi-group with 0 and 1, and over a finite field cannot be associative. The lattice of subvarieties of a variety of Lie algebras over a field is modular, but not distributive in general (cf. Modular lattice; Distributive lattice). The lattice is distributive only in the case of an infinite field. Bases of identities of specific Lie algebras have been found only in a few non-trivial cases: for ( or ), and also for some metabelian Lie algebras. Important results have been obtained concerning Lie algebras with the identity (see Lie algebra, nil).

References

[1]	V.A. Artamonov, "Lattices of varieties of linear algebras" Russian Math. Surveys , 33 : 2 (1978) pp. 155–193 Uspekhi Mat. Nauk , 33 : 2 (1978) pp. 135–167
[2]	R.K. Amayo, I. Stewart, "Infinite-dimensional Lie algebras" , Noordhoff (1974)
[3]	Yu.A. Bakhturin, "Lectures on Lie algebras" , Akademie Verlag (1978)
[4]	Yu.A. Bakhturin, "Identical relations in Lie algebras" , VNU , Utrecht (1987)