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 \mathfrak g has finite dimension, Levi's theorem states that this exact sequence splits; i.e., there exists a (necessarily semisimple) subalgebra of \mathfrak g 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.

Definition

Let k be a field and let be a finite-dimensional Lie algebra over k. 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 {0} 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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