Difference between revisions of "Lie algebra, reductive"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) {{MR|0218496}} {{ZBL|0132.27803}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.-P. Serre, "Algèbres de Lie semi-simples complexes" , Benjamin (1966) {{MR|0215886}} {{ZBL|0144.02105}} </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) {{MR|0682756}} {{ZBL|0319.17002}} </TD></TR></table> |
Revision as of 14:50, 24 March 2012
A finite-dimensional Lie algebra over a field of characteristic 0 whose adjoint representation is completely reducible (cf. Adjoint representation of a Lie group; Representation of a Lie algebra). The property that a Lie algebra
is reductive is equivalent to any of the following properties:
1) the radical of
coincides with the centre
;
2) , where
is a semi-simple ideal of
;
3) , where the
are prime ideals;
4) 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 .
An important class of reductive Lie algebras over are the compact Lie algebras (see Lie group, compact). A Lie group with a reductive Lie algebra is often called a reductive Lie group. A Lie algebra over
is reductive if and only if it is isomorphic to the Lie algebra of a reductive algebraic group over
.
A generalization of the concept of a reductive Lie algebra is the following. A subalgebra of a finite-dimensional Lie algebra
over
is said to be reductive in
if the adjoint representation
is completely reducible. In this case
is a reductive Lie algebra. If
is algebraically closed, then for a subalgebra
of
to be reductive it is necessary and sufficient that
consists of semi-simple linear transformations.
References
[1] | J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803 |
[2] | J.-P. Serre, "Algèbres de Lie semi-simples complexes" , Benjamin (1966) MR0215886 Zbl 0144.02105 |
[3] | N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002 |
Lie algebra, reductive. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_reductive&oldid=21884