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

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,

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

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