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| {{Lie groups |Algebras}}
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| In [[mathematics]], the '''Killing form''', named after [[Wilhelm Killing]], is a [[symmetric bilinear form]] that plays a basic role in the theories of [[Lie group]]s and [[Lie algebra]]s. The Killing form was essentially introduced into Lie algebra theory by {{harvs|txt |authorlilnk=Élie Cartan |first=Élie |last=Cartan |year=1894}} in his thesis; although Killing had previously made a passing mention of it, he made no serious use of it.
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| == Definition ==
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| Consider a [[Lie algebra]] '''g''' over a [[field (mathematics)|field]] ''K''. Every element ''x'' of '''g''' defines the [[adjoint endomorphism]] ad(''x'') (also written as ad<sub>''x''</sub>) of '''g''' with the help of the Lie bracket, as
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| :ad(''x'')(''y'') = [''x'', ''y''].
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| Now, supposing '''g''' is of finite dimension, the [[trace of a matrix|trace]] of the composition of two such endomorphisms defines a [[symmetric bilinear form]]
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| :''B''(''x'', ''y'') = trace(ad(''x'')ad(''y'')),
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| with values in ''K'', the '''Killing form''' on '''g'''.
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| == Properties ==
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| * The Killing form ''B'' is bilinear and symmetric.
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| * The Killing form is an invariant form, in the sense that it has the 'associativity' property
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| ::''B''([''x'',''y''], ''z'') = ''B''(''x'', [''y'', ''z'']),
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| : where [ , ] is the [[Lie algebra#Definition_and_first_properties|Lie bracket]].
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| * If '''g''' is a [[simple Lie algebra]] then any invariant symmetric bilinear form on '''g''' is a scalar multiple of the Killing form.
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| * The Killing form is also invariant under [[automorphism]]s ''s'' of the algebra '''g''', that is,
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| ::''B''(''s''(''x''), ''s''(''y'')) = ''B''(''x'', ''y'')
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| :for ''s'' in Aut('''g''').
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| * The [[Cartan criterion]] states that a Lie algebra is [[semisimple Lie algebra|semisimple]] if and only if the Killing form is [[degenerate form|non-degenerate]].
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| * The Killing form of a Lie algebra is identically zero iff it is a [[solvable Lie algebra]]. In particular, the Killing form of a [[nilpotent Lie algebra]] is identically zero.
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| * If ''I'', ''J'' are two [[ideal of a Lie algebra|ideals]] in a Lie algebra '''g''' with zero intersection, then ''I'' and ''J'' are [[orthogonal]] subspaces with respect to the Killing form.
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| * If a given Lie algebra '''g''' is a direct sum of its ideals ''I''<sub>1</sub>,...,''I<sub>n</sub>'', then the Killing form of '''g''' is the direct sum of the Killing forms of the individual summands.
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| == Matrix elements ==
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| Given a basis ''e<sup>i</sup>'' of the Lie algebra '''g''', the matrix elements of the Killing form are given by
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| :<math>B^{ij}= \mathrm{tr} (\mathrm{ad}(e^i)\circ \mathrm{ad}(e^j)) / I_{ad}</math> | |
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| where ''I''<sub>ad</sub> is the [[Dynkin index]] of the adjoint representation of '''g'''. Here
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| :<math>\left(\textrm{ad}(e^i) \circ \textrm{ad}(e^j)\right)(e^k)= [e^i, [e^j, e^k]] = {c^{im}}_{n} {c^{jk}}_{m} e^n </math>
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| in [[Einstein summation notation]] and so we can write
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| :<math>B^{ij} = \frac{1}{I_{{ad}}} {c^{im}}_{n} {c^{jn}}_{m}</math> | |
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| where the <math>c^{ij}_k</math> are the [[Algebra over a field#Structure coefficients|structure coefficient]]s of the Lie algebra. The Killing form is the simplest 2-[[tensor]] that can be formed from the structure constants.
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| In the above indexed definition, we are careful to distinguish upper and lower indexes (''co-'' and ''contra-variant'' indexes). This is because, in many cases, the Killing form can be used as a metric tensor on a manifold, in which case the distinction becomes an important one for the transformation properties of tensors. When the Lie algebra is semisimple, its Killing form is nondegenerate, and hence can be used as a [[metric tensor]] to raise and lower indexes. In this case, it is always possible to choose a basis for '''g''' such that the structure constants with all upper indexes are [[antisymmetric tensor|completely antisymmetric]].
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| The Killing form for some Lie algebras '''g''' are (for ''X'', ''Y'' in '''g'''):
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| {| class="wikitable"
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| |-
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| ! '''g''' || ''B''(''X'', ''Y'')
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| |-
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| | '''gl'''(''n'', '''R''') || 2''n'' tr(''XY'') − 2 tr(''X'')tr(''Y'')
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| |-
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| | '''sl'''(''n'', '''R''') || 2''n'' tr(''XY'')
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| |-
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| | '''su'''(''n'') || 2''n'' tr(''XY'')
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| |-
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| | '''so'''(''n'', '''R''') || (''n''−2) tr(''XY'')
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| |-
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| | '''so'''(''n'') || (''n''−2) tr(''XY'')
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| |-
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| | '''sp'''(''n'', '''R''') || (2''n''+2) tr(''XY'')
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| |-
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| | '''sp'''(''n'', '''C''') || (2''n''+2) tr(''XY'')
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| |}
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| == Connection with real forms ==
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| {{main|Real form (Lie theory)}}
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| Suppose that '''g''' is a [[semisimple Lie algebra]] over the field of real numbers. By Cartan's criterion, the Killing form is nondegenerate, and can be diagonalized in a suitable basis with the diagonal entries ±1. By [[Sylvester's law of inertia]], the number of positive entries is an invariant of the bilinear form, i.e. it does not depend on the choice of the diagonalizing basis, and is called the '''index''' of the Lie algebra '''g'''. This is a number between 0 and the dimension of '''g''' which is an important invariant of the real Lie algebra. In particular, a real Lie algebra '''g''' is called '''compact''' if the Killing form is [[negative definite]]. It is known that under the [[Lie correspondence]], [[compact Lie algebra]]s correspond to [[compact Lie group]]s.
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| If '''g'''<sub>'''C'''</sub> is a semisimple Lie algebra over the complex numbers, then there are several non-isomorphic real Lie algebras whose [[complexification]] is '''g'''<sub>'''C'''</sub>, which are called its '''real forms'''. It turns out that every complex semisimple Lie algebra admits a unique (up to isomorphism) compact real form '''g'''. The real forms of a given complex semisimple Lie algebra are frequently labeled by the positive index of inertia of their Killing form.
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| For example, the complex [[special linear group|special linear algebra]] '''sl'''(2, '''C''') has two real forms, the real special linear algebra, denoted '''sl'''(2, '''R'''), and the [[special unitary group|special unitary algebra]], denoted '''su'''(2). The first one is noncompact, the so-called '''split real form''', and its Killing form has signature (2,1). The second one is the compact real form and its Killing form is negative definite, i.e. has signature (0,3). The corresponding Lie groups are the noncompact group SL(2, '''R''') of 2 × 2 real matrices with the unit determinant and the special unitary group [[SU(2)]], which is compact.
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| == See also ==
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| * [[Casimir invariant]]
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| == References ==
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| *Daniel Bump, ''Lie Groups'' (2004), Graduate Texts In Mathematics, '''225''', Springer-Verlag. ISBN 978-0-387-21154-1
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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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| *Jurgen Fuchs, ''Affine Lie Algebras and Quantum Groups'', (1992) Cambridge University Press. ISBN 0-521-48412-X
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| *{{Fulton-Harris}}
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| *{{springer|title=Killing form|id=p/k055400}}
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| [[Category:Lie groups]]
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| [[Category:Lie algebras]]
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