# 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)
How to Cite This Entry:
Lie algebras, variety of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebras,_variety_of&oldid=47627
This article was adapted from an original article by Yu.A. Bakhturin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article