Lie algebra, algebraic
The Lie algebra of an algebraic subgroup (see Algebraic group) of the general linear group of all automorphisms of a finite-dimensional vector space over a field
. If
is an arbitrary subalgebra of the Lie algebra of all endomorphisms of
, there is a smallest algebraic Lie algebra containing
; it is called the algebraic envelope (or hull) of the Lie subalgebra
. For a Lie algebra
over an arbitrary algebraically closed field
to be algebraic it is necessary that together with every linear operator
its semi-simple and nilpotent components
and
should lie in
(see Jordan decomposition). This condition determines the so-called almost-algebraic Lie algebras. However, it is not sufficient in order that
be an algebraic Lie algebra. In the case of a field
of characteristic 0, a necessary and sufficient condition for a Lie algebra
to be algebraic is that, together with
and
, all operators of the form
should lie in
, where
is an arbitrary
-linear mapping from
into
. The structure of an algebraic algebra has been investigated
in the case of a field of characteristic .
A Lie algebra over a commutative ring
in which for any element
the adjoint transformation
defined on
is the root of some polynomial with leading coefficient 1 and remaining coefficients from
. A finite-dimensional Lie algebra over a field
is an algebraic Lie algebra. The converse is false: Over any field
there are infinite-dimensional algebraic Lie algebras with finitely many generators [4]. A number of questions about algebraic Lie algebras have been solved in the class of nil Lie algebras (cf. Lie algebra, nil).
References
[1] | A. Borel, "Linear algebraic groups" , Benjamin (1969) |
[2] | C. Chevalley, "Théorie des groupes de Lie" , 2 , Hermann (1951) |
[3] | G.B. Seligman, "Modular Lie algebras" , Springer (1967) |
[4] | E.S. Golod, "On nil algebras and residually finite groups" Izv. Akad. Nauk SSSR Ser. Mat. , 28 : 2 (1964) pp. 273–276 (In Russian) |
Comments
References
[a1] | G.P. Hochschild, "Basic theory of algebraic groups and Lie algebras" , Springer (1981) |
Lie algebra, algebraic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_algebraic&oldid=14719