Killing form
A certain bilinear form on a finite-dimensional Lie algebra, introduced by W. Killing . Let be a finite-dimensional Lie algebra over a field
. By the Killing form on
is meant the bilinear form
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where denotes the trace of a linear operator, and
is the image of
under the adjoint representation of
(cf. also Adjoint representation of a Lie group), i.e. the linear operator on the vector space
defined by the rule
, where
is the commutation operator in the Lie algebra
. The Killing form is symmetric. The operators
,
, are skew-symmetric with respect to the Killing form, that is,
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If is an ideal of
, then the restriction of the Killing form to
is the same as the Killing form of
. Each commutative ideal
is contained in the kernel of the Killing form. If the Killing form is non-degenerate, then the algebra
is semi-simple (cf. Lie algebra, semi-simple).
Suppose that the characteristic of the field is 0. Then the radical of
is the same as the orthocomplement with respect to the Killing form of the derived subalgebra
. The algebra
is solvable (cf. Lie algebra, solvable) if and only if
, i.e. when
for all
(Cartan's solvability criterion). If
is nilpotent (cf. Lie algebra, nilpotent), then
for all
. The algebra
is semi-simple if and only if the Killing form is non-degenerate (Cartan's semi-simplicity criterion).
Every complex semi-simple Lie algebra contains a real form (the compact Weyl form, see Complexification of a Lie algebra) on which the Killing form is negative definite.
References
[1a] | W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen I" Math. Ann. , 31 (1888) pp. 252–290 |
[1b] | W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen II" Math. Ann. , 33 (1889) pp. 1–48 |
[1c] | W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen III" Math. Ann. , 34 (1889) pp. 57–122 |
[1d] | W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen IV" Math. Ann. , 36 (1890) pp. 161–189 |
[2] | E. Cartan, "Sur la structure des groupes de transformations finis et continus" , Oevres Complètes , 1 , CNRS (1984) pp. 137–288 |
[3] | N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) |
[4] | I. Kaplansky, "Lie algebras and locally compact groups" , Chicago Univ. Press (1971) |
[5] | M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) |
Comments
The Killing form is a key tool in the Killing–Cartan classification of semi-simple Lie algebras over fields of characteristic 0. If
, the Killing form on a semi-simple Lie algebra may be degenerate.
The Killing form is also called the Cartan–Killing form.
Let be a basis for the Lie algebra
, and let the corresponding structure constants be
, so that
(summation convention). Then in terms of these structure constants the Killing form is given by
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The metric (tensor) is called the Cartan metric, especially in the theoretical physics literature. Using
one can lower indices (cf. Tensor on a vector space) to obtain "structure constants"
which are completely anti-symmetric in their indices. (A direct consequence of the Jacobi identity and equivalent to the anti-symmetry of the operator
with respect to
; cf. above.)
References
[a1] | L. O'Raifeartaigh, "Group structure of gauge theories" , Cambridge Univ. Press (1986) |
[a2] | V.S. Varadarajan, "Lie groups, Lie algebras and their representations" , Springer, reprint (1984) |
[a3] | J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) pp. §5.4 |
Killing form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Killing_form&oldid=26775