Lie algebras, variety of

over a ring $ k $

A class $ \mathfrak V $ of Lie algebras (cf. Lie algebra) over $ k $ that satisfy a fixed system of identities. The most prevalent varieties of Lie algebras are the following: the variety $ \mathfrak A $ of Abelian Lie algebras specified by the identity $ [ x , y] \equiv 0 $, the variety $ \mathfrak N _ {c} $ of nilpotent Lie algebras of class $ c $( in which any products of length greater than $ c $ are equal to zero), the variety $ \mathfrak S _ {l} $ of solvable Lie algebras of length $ \leq l $( in which the derived series converges to zero in no more than $ l $ steps). The totality $ v ( k) $ of all varieties of Lie algebras over $ k $ is a groupoid with respect to multiplication: $ \mathfrak W = \mathfrak U \mathfrak V $, where $ \mathfrak W $ is the class of extensions of algebras from $ \mathfrak V $ by means of ideals from $ \mathfrak U $; $ \mathfrak S _ {l} = \mathfrak A ^ {l} $; the algebras of $ \mathfrak A ^ {2} $ 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 $ k $ is a Noetherian ring). If $ k $ is a field of characteristic $ p > 0 $, there are examples of locally finite varieties of Lie algebras lying in $ \mathfrak A ^ {3} $ and not having a finite basis of identities. In the case of a field $ k $ 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 $ \mathfrak N _ {c} \mathfrak A \cap \mathfrak A \mathfrak N _ {c} $ over any Noetherian ring, $ \mathfrak N _ {c} \mathfrak A \cap \mathfrak N _ {2} \mathfrak N _ {c} $ over any field of characteristic $ \neq 2 $, and $ \mathop{\rm var} ( k _ {2} ) $, defined by identities that are true in the Lie algebra $ k _ {2} $ of matrices of order 2 over a field $ k $ with $ \mathop{\rm char} ( k) = 0 $. Over a field $ k $ of characteristic 0 there are still no examples of a finite-dimensional Lie algebra $ A $ such that $ \mathop{\rm var} ( A) $ is infinitely based, but there are such examples over an infinite field $ k $ of characteristic $ p > 0 $. Over a finite field, or, more generally, over any finite ring $ k $ with a unit, the identities of a finite Lie algebra $ A $ follow from a finite subsystem of them.

A variety of Lie algebras $ \mathop{\rm var} ( A) $ generated by a finite algebra $ A $ is called a Cross variety and is contained in a Cross variety $ \mathfrak C ( f , m , c ) $ consisting of Lie algebras in which all principal factors have order $ \leq m $, all nilpotent factors have class $ \leq c $ and all inner derivations $ \mathop{\rm ad} x $ are annihilated by a unitary polynomial $ f \in k [ t] $. 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 $ v ( k) $ over an infinite field is a free semi-group with 0 and 1, and over a finite field $ v ( k) $ cannot be associative. The lattice $ {\mathcal L} ( \mathfrak V ) $ of subvarieties of a variety of Lie algebras $ \mathfrak V $ over a field $ k $ is modular, but not distributive in general (cf. Modular lattice; Distributive lattice). The lattice $ {\mathcal L} ( \mathfrak A ^ {2} ) $ 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 $ k _ {2} $( $ \mathop{\rm char} ( k) = 0 $ or $ \mathop{\rm char} ( k) = 2 $), and also for some metabelian Lie algebras. Important results have been obtained concerning Lie algebras with the identity $ ( \mathop{\rm ad} x ) ^ {n} = 0 $( 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)