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 $\le l$( 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 $\ne 2$, and $\mathrm{var}(k_2)$, defined by identities that are true in the Lie algebra $k_2$ of matrices of order 2 over a field $k$ with $\mathrm{char}(k)=0$. Over a field $k$ of characteristic 0 there are still no examples of a finite-dimensional Lie algebra $A$ such that $\mathrm{var}(A)$ 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 $\mathrm{var}(A)$ 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 $\le m$, all nilpotent factors have class $\le c$ and all inner derivations $\mathrm{ad}x$ 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 $k_2$( $\mathrm{char}(k)=0$ or $\mathrm{char}(k)=2$), and also for some metabelian Lie algebras. Important results have been obtained concerning Lie algebras with the identity $(\mathrm{ad}x{)}^n=0$( see Lie algebra, nil).

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