In the mathematical field of Lie theory, the radical of a Lie algebra
is the largest solvable ideal of
[1]
The radical, denoted by
, fits into the exact sequence
.
where
is semisimple. When the ground field has characteristic zero and
has finite dimension, Levi's theorem states that this exact sequence splits; i.e., there exists a (necessarily semisimple) subalgebra of
that is isomorphic to the semisimple quotient
via the restriction of the quotient map
A similar notion is a Borel subalgebra, which is a (not necessarily unique) maximal solvable subalgebra.
- ^ Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2010), Algebras, Rings and Modules: Lie Algebras and Hopf Algebras, Mathematical Surveys and Monographs, vol. 168, Providence, RI: American Mathematical Society, p. 15, doi:10.1090/surv/168, ISBN 978-0-8218-5262-0, MR 2724822.