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| In mathematics, '''Cartan's criterion''' gives conditions for a [[Lie algebra]] in characteristic 0 to be [[solvable Lie algebra|solvable]], which implies a related criterion for the Lie algebra to be [[semisimple Lie algebra|semisimple]]. It is based on the notion of the [[Killing form]], a [[symmetric bilinear form]] on <math>\mathfrak{g}</math> defined by the formula
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| : <math>K(u,v)=\operatorname{tr}(\operatorname{ad}(u)\operatorname{ad}(v)),</math> | |
| where tr denotes the [[Trace (linear algebra)|trace of a linear operator]]. The criterion was introduced by {{harvs|txt|authorlink=Élie Cartan|first=Élie|last= Cartan|year=1894}}.
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| == Cartan's criterion for solvability ==
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| Cartan's criterion for solvability states:
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| :''A Lie subalgebra <math>\mathfrak{g}</math> of endomorphisms of a finite dimensional vector space over a [[field (mathematics)|field]] of [[characteristic zero]] is solvable if and only if <math>Tr(ab)=0</math> whenever <math>a\in\mathfrak{g},b\in[\mathfrak{g},\mathfrak{g}].</math>''
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| The fact that <math>Tr(ab)=0</math> in the solvable case follows immediately from [[Lie–Kolchin theorem|Lie's theorem]] that solvable Lie algebras in characteristic 0 can be put in upper triangular form.
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| Applying Cartan's criterion to the adjoint representation gives:
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| :''A finite-dimensional Lie algebra <math>\mathfrak{g}</math> over a [[field (mathematics)|field]] of [[characteristic zero]] is solvable if and only if <math>K(\mathfrak{g},[\mathfrak{g},\mathfrak{g}])=0</math> (where K is the Killing form).''
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| == Cartan's criterion for semisimplicity ==
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| Cartan's criterion for semisimplicity states:
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| : ''A finite-dimensional Lie algebra <math>\mathfrak{g}</math> over a [[field (mathematics)|field]] of [[characteristic zero]] is semisimple if and only if the Killing form is [[degenerate form|non-degenerate]].''
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| {{harvtxt|Dieudonné|1953}} gave a very short proof that if a finite dimensional Lie algebra (in any characteristic) has a [[Quadratic Lie algebra|non-degenerate invariant bilinear form]] and no non-zero abelian ideals, and in particular if its Killing form is non-degenerate, then it is a sum of simple Lie algebras.
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| Conversely, it follows easily from Cartan's criterion for solvability that a semisimple algebra (in characteristic 0) has a non-degenerate Killing form.
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| ==Examples==
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| Cartan's criteria fail in characteristic ''p''>0; for example:
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| *the Lie algebra SL<sub>''p''</sub>(''k'') is simple if ''k'' has characteristic not 2 and has vanishing Killing form, though it does have a nonzero invariant bilinear form given by (''a'',''b'') = Tr(''ab'').
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| *the Lie algebra with basis ''a''<sub>''n''</sub> for ''n''∈'''Z'''/''p'''''Z''' and bracket [''a''<sub>''i''</sub>,''a''<sub>''j''</sub>] = (''i''−''j'')''a''<sub>''i''+''j''</sub> is simple for ''p''>2 but has no nonzero invariant bilinear form.
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| *If ''k'' has characteristic 2 then the semidirect product gl<sub>2</sub>(''k'').''k''<sup>2</sup> is a solvable Lie algebra, but the Killing form is not identically zero on its derived algebra sl<sub>2</sub>(''k'').''k''<sup>2</sup>.
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| If a finite dimensional Lie algebra is nilpotent, then the Killing form is identically zero (and more generally the Killing form vanishes on any nilpotent ideal). The converse is false: there are non-nilpotent Lie algebras whose Killing form vanishes. An example is given by the semidirect product of an abelian Lie algebra ''V'' with a 1-dimensional Lie algebra acting on ''V'' as an endomorphism ''b'' such that ''b'' is not nilpotent and Tr(''b''<sup>2</sup>)=0.
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| In characteristic 0, every reductive Lie algebra (one that is a sum of abelian and simple Lie algebras) has a non-degenerate invariant symmetric bilinear form. However the converse is false: a Lie algebra with a non-degenerate invariant symmetric bilinear form need not be a sum of simple and abelian Lie algebras. A typical counterexample is ''G'' = ''L''[''t'']/''t''<sup>''n''</sup>''L''[''t''] where ''n''>1, ''L'' is a simple complex Lie algebra with a bilinear form (,), and the bilinear form on ''G'' is given by taking the coefficient of ''t''<sup>''n''−1</sup> of the '''C'''[''t'']-valued bilinear form on ''G'' induced by the form on ''L''. The bilinear form is non-degenerate, but the Lie algebra is not a sum of simple and abelian Lie algebras.
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| == References ==
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| *{{Citation | last1=Cartan | first1=Élie | title=Sur la structure des groupes de transformations finis et continus | url=http://books.google.com/books?id=JY8LAAAAYAAJ | publisher=Nony | series=Thesis | year=1894}}
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| *{{Citation | last1=Dieudonné | first1=Jean | author1-link=Jean Dieudonné | title=On semi-simple Lie algebras | jstor=2031832 | mr=0059262 | year=1953 | journal=[[Proceedings of the American Mathematical Society]] | issn=0002-9939 | volume=4 | pages=931–932}}
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| *{{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Lie algebras and Lie groups | origyear=1964 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-55008-2 | doi=10.1007/978-3-540-70634-2 | mr=2179691 | year=2006 | volume=1500}}
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| == See also ==
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| * [[Modular Lie algebra]]
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| [[Category:Lie algebras]]
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