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)

References

[a1]	G.P. Hochschild, "Basic theory of algebraic groups and Lie algebras" , Springer (1981)