View source for Lie algebra, algebraic

The Lie algebra of an algebraic subgroup (see
[[Algebraic group|Algebraic group]]) of the
[[General linear group|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|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
[[#References|[4]]]. A number of questions about algebraic Lie
algebras have been solved in the class of nil Lie algebras (cf.
[[Lie algebra, nil|Lie algebra, nil]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD
valign="top"> A. Borel, "Linear algebraic groups" , Benjamin
(1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">
C. Chevalley, "Théorie des groupes de Lie" , '''2''' , Hermann
(1951) {{MR|0051242}} {{ZBL|0054.01303}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">
G.B. Seligman, "Modular Lie algebras" , Springer
(1967) {{MR|0245627}} {{ZBL|0189.03201}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">
E.S. Golod, "On nil algebras and residually finite groups"
''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''28''' : 2 (1964) pp. 273–276
(In Russian) {{MR|}} {{ZBL|}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD
valign="top"> G.P. Hochschild, "Basic theory of algebraic groups and
Lie algebras" , Springer (1981) {{MR|0620024}} {{ZBL|0589.20025}} </TD></TR></table>