Radical of a Lie algebra


In the mathematical field of Lie theory, the radical of a Lie algebra is the largest solvable ideal of
The radical, denoted by, fits into the exact sequence
where is semisimple. When the ground field has characteristic zero and has finite dimension, then Levi's theorem states that this exact sequence splits; i.e., there exists a subalgebra of that is isomorphic to the semisimple quotient via the quotient map
A similar notion is a Borel subalgebra, which is a maximal solvable subalgebra.

Definition

Let be a field and let be a finite-dimensional Lie algebra over. There exists a unique maximal solvable ideal, called the radical, 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. Now consider the sum of all the solvable ideals of. It is nonempty since is a solvable ideal, and it is a solvable ideal by the sum property just derived. Clearly it is the unique maximal solvable ideal.

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