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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>
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| == Definition ==
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| 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.
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| 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.
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| == Related concepts ==
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| * A Lie algebra is [[Semisimple Lie algebra|semisimple]] if and only if its radical is <math>0</math>.
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| * A Lie algebra is [[Reductive Lie algebra|reductive]] if and only if its radical equals its center.
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| {{algebra-stub}}
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| [[Category:Lie algebras]]
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Latest revision as of 23:02, 21 April 2014
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