Radical of a Lie algebra

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In the mathematical field of Lie theory, the radical of a Lie algebra is the largest solvable ideal of [1]

Definition[edit]

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.

Related concepts[edit]

  • 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.

References[edit]

  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 .