Radical of a Lie algebra

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In the mathematical field of Lie theory, the radical of a Lie algebra \mathfrak{g} is the largest solvable ideal of \mathfrak{g}.[1]


Let k be a field and let \mathfrak{g} be a finite-dimensional Lie algebra over k. There exists a unique maximal solvable ideal, called the radical, for the following reason.

Firstly let \mathfrak{a} and \mathfrak{b} be two solvable ideals of \mathfrak{g}. Then \mathfrak{a}+\mathfrak{b} is again an ideal of \mathfrak{g}, and it is solvable because it is an extension of (\mathfrak{a}+\mathfrak{b})/\mathfrak{a}\simeq\mathfrak{b}/(\mathfrak{a}\cap\mathfrak{b}) by \mathfrak{a}. Now consider the sum of all the solvable ideals of \mathfrak{g}. 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.

Related concepts[edit]

  • A Lie algebra is semisimple if and only if its radical is 0.
  • A Lie algebra is reductive if and only if its radical equals its center.


  1. ^ 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 .