View source for Lie algebra, reductive

A finite-dimensional [[Lie algebra|Lie algebra]] over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l0585001.png" /> of characteristic 0 whose adjoint representation is completely reducible (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]; [[Representation of a Lie algebra|Representation of a Lie algebra]]). The property that a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l0585002.png" /> is reductive is equivalent to any of the following properties:

1) the radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l0585003.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l0585004.png" /> coincides with the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l0585005.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l0585006.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l0585007.png" /> is a semi-simple ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l0585008.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l0585009.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l05850010.png" /> are prime ideals;

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l05850011.png" /> admits a faithful completely-reducible finite-dimensional linear representation.

The property that a Lie algebra is reductive is preserved by both extension and restriction of the ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l05850012.png" />.

An important class of reductive Lie algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l05850013.png" /> are the compact Lie algebras (see [[Lie group, compact|Lie group, compact]]). A Lie group with a reductive Lie algebra is often called a reductive Lie group. A Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l05850014.png" /> is reductive if and only if it is isomorphic to the Lie algebra of a reductive algebraic group over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l05850015.png" />.

A generalization of the concept of a reductive Lie algebra is the following. A subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l05850016.png" /> of a finite-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l05850017.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l05850018.png" /> is said to be reductive in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l05850019.png" /> if the adjoint representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l05850020.png" /> is completely reducible. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l05850021.png" /> is a reductive Lie algebra. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l05850022.png" /> is algebraically closed, then for a subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l05850023.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l05850024.png" /> to be reductive it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058500/l05850025.png" /> consists of semi-simple linear transformations.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.-P. Serre,   "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,   "Algèbres de Lie semi-simples complexes" , Benjamin  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR></table>