Radical of a Lie algebra
In the mathematical field of Lie theory, the radical of a Lie algebra is the largest solvable ideal of
[1]
Definition
Let be a field and let
be a finite-dimensional Lie algebra over
. A maximal solvable ideal, which is called the radical, exists for the following reason.
Firstly let and
be two solvable ideals of
. Then
is again an ideal of
, and it is solvable because it is an extension of
by
. Therefore we may also define the radical of
as the sum of all the solvable ideals of
, hence the radical of
is unique. Secondly, as
is always a solvable ideal of
, the radical of
always exists.
Related concepts
- A Lie algebra is semisimple if and only if its radical is
.
- A Lie algebra is reductive if and only if its radical equals its center.
References
- ↑ Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2010), Algebras, Rings and Modules: Lie Algebras and Hopf Algebras, Mathematical Surveys and Monographs 168, Providence, RI: American Mathematical Society, p. 15, doi:10.1090/surv/168, ISBN 978-0-8218-5262-0, MR 2724822.