Killing form

2020 Mathematics Subject Classification: Primary: 17B [MSN][ZBL]

The Killing form is a certain bilinear form on a finite-dimensional Lie algebra, introduced by W. Killing . Let $\def\f#1{\mathfrak{#1}}\f G$ be a finite-dimensional Lie algebra over a field $k$. By the Killing form on $\f G$ is meant the bilinear form

$$\def\tr{\textrm{tr}\;}\def\ad{\textrm{ad}\;}B(x,y) = \tr(\ad x \cdot \ad y),\quad x,y\in \f G $$ where $\tr$ denotes the trace of a linear operator, and $\ad x$ is the image of $x$ under the adjoint representation of $\f G$ (cf. also Adjoint representation of a Lie group), i.e. the linear operator on the vector space $\f G$ defined by the rule $z\mapsto [z,x]$, where $[\;,\;]$ is the commutation operator in the Lie algebra $\f G$. The Killing form is symmetric. The operators $\ad x$, $x\in \f G$, are skew-symmetric with respect to the Killing form, that is,

$$B([x,y],z) = B(x,[y,z])\quad \textrm{ for all } x,y,z\in \f G.$$ If $\f G_0$ is an ideal of $\f G$, then the restriction of the Killing form to $\f G_0$ is the same as the Killing form of $\f G_0$. Each commutative ideal $\f G_0$ is contained in the kernel of the Killing form. If the Killing form is non-degenerate, then the algebra $\f G$ is semi-simple (cf. Lie algebra, semi-simple).

Suppose that the characteristic of the field $k$ is 0. Then the radical of $\f G$ is the same as the orthocomplement with respect to the Killing form of the derived subalgebra $\f G' = [\f G,\f G]$. The algebra $\f G$ is solvable (cf. Lie algebra, solvable) if and only if $\f G\perp \f G'$, i.e. when $B([x,y],z) = 0$ for all $x,y,z\in \f G$ (Cartan's solvability criterion). If $\f G$ is nilpotent (cf. Lie algebra, nilpotent), then $B(x,y) = 0$ for all $x,y\in\f G$. The algebra $\f G$ 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 $\Gamma$ (the compact Weyl form, see Complexification of a Lie algebra) on which the Killing form is negative definite.

Comments

The Killing form is a key tool in the Killing–Cartan classification of semi-simple Lie algebras over fields $k$ of characteristic 0. If $\textrm{char}\; k \ne 0$, the Killing form on a semi-simple Lie algebra may be degenerate.

The Killing form is also called the Cartan–Killing form.

Let $X_1,\dots,X_n$ be a basis for the Lie algebra $L_1$, and let the corresponding structure constants be $\def\g{\gamma}\g_{ij}^k$, so that $[X_i,X_j] = \g_{ij}^k X_k$ (summation convention). Then in terms of these structure constants the Killing form is given by

$$B(X_a,X_b) = g_{ab} = \g_{ac}^d\g_{bd}^c$$ The metric (tensor) $g_{ab}$ is called the Cartan metric, especially in the theoretical physics literature. Using $g_{ab}$ one can lower indices (cf. Tensor on a vector space) to obtain "structure constants" $\g_{abc} = g_{da} \g_{bc}^d$ which are completely anti-symmetric in their indices. (A direct consequence of the Jacobi identity and equivalent to the anti-symmetry of the operator $\ad y$ with respect to $B(x,z)$; cf. above.)

References

[Bo]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras", Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002
[Ca]	E. Cartan, "Sur la structure des groupes de transformations finis et continus", Oevres Complètes, 1, CNRS (1984) pp. 137–288 Zbl 0007.10204 JFM Zbl 59.0430.02 JFM Zbl 25.0638.02
[Hu]	J.E. Humphreys, "Introduction to Lie algebras and representation theory", Springer (1972) pp. §5.4 MR0323842 Zbl 0254.17004
[Ka]	I. Kaplansky, "Lie algebras and locally compact groups", Chicago Univ. Press (1971) MR0276398 Zbl 0223.17001
[Ki]	W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen I" Math. Ann., 31 (1888) pp. 252–290 JFM Zbl 20.0368.03
[Ki2]	W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen II" Math. Ann., 33 (1889) pp. 1–48 JFM Zbl 20.0368.03
[Ki3]	W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen III" Math. Ann., 34 (1889) pp. 57–122 JFM Zbl 21.0376.01
[Ki4]	W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen IV" Math. Ann., 36 (1890) pp. 161–189
[Na]	M.A. Naimark, "Theory of group representations", Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018
[Va]	V.S. Varadarajan, "Lie groups, Lie algebras and their representations", Springer, reprint (1984) MR0746308 Zbl 0955.22500