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 $s=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(s_1,\dots ,s_m)$, all operators of the form $\mathrm{\Phi }(s)=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\mathrm{\Phi }(s_1),\dots ,\mathrm{\Phi }(s_m))$ should lie in $\mathfrak{g}$, where $\mathrm{\Phi }$ 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 $\mathrm{a}\mathrm{d}x:y\to [x,y]$ 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).

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