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In the [[mathematical]] field of [[Lie theory]], the '''radical''' of a [[Lie algebra]] <math>\mathfrak{g}</math> is the largest [[Solvable Lie algebra|solvable]] [[Ideal (Lie algebra)|ideal]] of <math>\mathfrak{g}.</math> | |||
== Definition == | |||
Let <math>k</math> be a field and let <math>\mathfrak{g}</math> be a finite-dimensional [[Lie algebra]] over <math>k</math>. A maximal solvable ideal, which is called the ''radical,'' exists for the following reason. | |||
Firstly let <math>\mathfrak{a}</math> and <math>\mathfrak{b}</math> be two solvable ideals of <math>\mathfrak{g}</math>. Then <math>\mathfrak{a}+\mathfrak{b}</math> is again an ideal of <math>\mathfrak{g}</math>, and it is solvable because it is an extension of <math>(\mathfrak{a}+\mathfrak{b})/\mathfrak{a}\simeq\mathfrak{b}/(\mathfrak{a}\cap\mathfrak{b})</math> by <math>\mathfrak{a}</math>. Therefore we may also define the radical of <math>\mathfrak{g}</math> as the sum of all the solvable ideals of <math>\mathfrak{g}</math>, hence the radical of <math>\mathfrak{g}</math> is unique. Secondly, as <math>\{0\}</math> is always a solvable ideal of <math>\mathfrak{g}</math>, the radical of <math>\mathfrak{g}</math> always exists. | |||
== Related concepts == | |||
* A Lie algebra is [[Semisimple Lie algebra|semisimple]] if and only if its radical is <math>0</math>. | |||
* A Lie algebra is [[Reductive Lie algebra|reductive]] if and only if its radical equals its center. | |||
{{algebra-stub}} | |||
[[Category:Lie algebras]] | |||
Revision as of 20:35, 20 January 2014
In the mathematical field of Lie theory, the radical of a Lie algebra is the largest solvable ideal of
Definition
Let be a field and let be a finite-dimensional Lie algebra over . A maximal solvable ideal, which is called the radical, exists 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 . Therefore we may also define the radical of as the sum of all the solvable ideals of , hence the radical of is unique. Secondly, as is always a solvable ideal of , the radical of always exists.
Related concepts
- 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.