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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583801.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583802.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583803.png" /> is an arbitrary subalgebra of the Lie algebra of all endomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583804.png" />, there is a smallest algebraic Lie algebra containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583805.png" />; it is called the algebraic envelope (or hull) of the Lie subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583806.png" />. For a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583807.png" /> over an arbitrary algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583808.png" /> to be algebraic it is necessary that together with every linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l0583809.png" /> its semi-simple and nilpotent components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838011.png" /> should lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838012.png" /> (see [[Jordan decomposition|Jordan decomposition]]). This condition determines the so-called almost-algebraic Lie algebras. However, it is not sufficient in order that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838013.png" /> be an algebraic Lie algebra. In the case of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838014.png" /> of characteristic 0, a necessary and sufficient condition for a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838015.png" /> to be algebraic is that, together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838017.png" />, all operators of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838018.png" /> should lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838020.png" /> is an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838021.png" />-linear mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838022.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838023.png" />. The structure of an algebraic algebra has been investigated

in the case of a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838024.png" />.

A Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838025.png" /> over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838026.png" /> in which for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838027.png" /> the adjoint transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838028.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838029.png" /> is the root of some polynomial with leading coefficient 1 and remaining coefficients from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838030.png" />. A finite-dimensional Lie algebra over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838031.png" /> is an algebraic Lie algebra. The converse is false: Over any field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058380/l05838032.png" /> 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)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Chevalley,   "Théorie des groupes de Lie" , '''2''' , Hermann  (1951)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.B. Seligman,   "Modular Lie algebras" , Springer  (1967)</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)</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)</TD></TR></table>