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 $V$ over a field $k$. If $\mathfrak g$ is an arbitrary subalgebra of the Lie algebra of all endomorphisms of $V$, there is a smallest algebraic Lie algebra containing $\mathfrak g$; it is called the algebraic envelope (or hull) of the Lie subalgebra $\mathfrak g$. For a Lie algebra $\mathfrak g$ over an arbitrary algebraically closed field $k$ to be algebraic it is necessary that together with every linear operator $u\in{\mathfrak g}$ its semi-simple and nilpotent components $s$ and $n$ should lie in ${\mathfrak g}$ (see Jordan decomposition). This condition determines the so-called almost-algebraic Lie algebras. However, it is not sufficient in order that ${\mathfrak g}$ be an algebraic Lie algebra. In the case of a field $k$ of characteristic 0, a necessary and sufficient condition for a Lie algebra ${\mathfrak g}$ to be algebraic is that, together with $n$ and $s={\rm diag}(s_1,\dots,s_m)$, all operators of the form $\Phi(s) = {\rm diag}(\Phi(s_1),\dots,\Phi(s_m))$ should lie in ${\mathfrak g}$, where $\Phi$ is an arbitrary $\mathbb Q$-linear mapping from $k$ into $k$. The structure of an algebraic algebra has been investigated in the case of a field of characteristic $p>0$.

A Lie algebra $L$ over a commutative ring $k$ in which for any element $x\in L$ the adjoint transformation ${\rm ad}\; x:y\to [x,y]$ defined on $L$ is the root of some polynomial with leading coefficient 1 and remaining coefficients from $k$. A finite-dimensional Lie algebra over a field $k$ is an algebraic Lie algebra. The converse is false: Over any field $k$ 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) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2]	C. Chevalley, "Théorie des groupes de Lie" , 2 , Hermann (1951) MR0051242 Zbl 0054.01303
[3]	G.B. Seligman, "Modular Lie algebras" , Springer (1967) MR0245627 Zbl 0189.03201
[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) MR0620024 Zbl 0589.20025