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) |
Lie algebras, variety of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebras,_variety_of&oldid=12482