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]

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.

Definition

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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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  • 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.

See also

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References

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  1. ^ 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.