Killing form

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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 groups and Lie algebras. The Killing form was essentially introduced into Lie algebra theory by Élie Cartan (1894) in his thesis; although Killing had previously made a passing mention of it, he made no serious use of it.

Definition[edit]

Consider a Lie algebra g over a field K. Every element x of g defines the adjoint endomorphism ad(x) (also written as adx) of g with the help of the Lie bracket, as

\mathrm{ad}(x)(y) = [x, y].

Now, supposing g is of finite dimension, the trace of the composition of two such endomorphisms defines a symmetric bilinear form

B(x, y) = \mathrm{trace}(\mathrm{ad}(x)\mathrm{ad}(y)),

with values in K, the Killing form on g.

Properties[edit]

  • The Killing form B is bilinear and symmetric.
  • The Killing form is an invariant form, in the sense that it has the 'associativity' property
B([x, y], z) = B(x, [y, z]),
where [ , ] is the Lie bracket.
  • If g is a simple Lie algebra then any invariant symmetric bilinear form on g is a scalar multiple of the Killing form.
  • The Killing form is also invariant under automorphisms s of the algebra g, that is,
B(s(x), s(y)) = B(x, y)
for s in Aut(g).
  • The Cartan criterion states that a Lie algebra is semisimple if and only if the Killing form is non-degenerate.
  • The Killing form of a nilpotent Lie algebra is identically zero.
  • If I, J are two ideals in a Lie algebra g with zero intersection, then I and J are orthogonal subspaces with respect to the Killing form.
  • The orthogonal complement with respect to B of an ideal is again an ideal.[1]
  • If a given Lie algebra g is a direct sum of its ideals I1,...,In, then the Killing form of g is the direct sum of the Killing forms of the individual summands.

Matrix elements[edit]

Given a basis ei of the Lie algebra g, the matrix elements of the Killing form are given by

B^{ij}= \mathrm{tr} (\mathrm{ad}(e^i)\circ \mathrm{ad}(e^j)) / I_{ad}

where Iad is the Dynkin index of the adjoint representation of g. Here

\left(\textrm{ad}(e^i) \circ \textrm{ad}(e^j)\right)(e^k)=  [e^i, [e^j, e^k]] = [e^i, {c^{jk}}_{m}e^m] = {c^{im}}_{n} {c^{jk}}_{m} e^n

in Einstein summation notation, where the cijk are the structure coefficients of the Lie algebra. The index k functions as column index and the index n as row index in the matrix ad(ei)ad(ej). Taking the trace amounts to putting k = n and summing, and so we can write

B^{ij} = \frac{1}{I_{{ad}}} {c^{im}}_{n} {c^{jn}}_{m}

The Killing form is the simplest 2-tensor that can be formed from the structure constants.

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 completely antisymmetric.

The Killing form for some Lie algebras g are (for X, Y in g):

g B(X, Y)
gl(n, R) 2n tr(XY) − 2 tr(X)tr(Y)
sl(n, R) 2n tr(XY)
su(n) 2n tr(XY)
so(n, R) (n−2) tr(XY)
so(n) (n−2) tr(XY)
sp(n, R) (2n+2) tr(XY)
sp(n, C) (2n+2) tr(XY)

Connection with real forms[edit]

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 algebras correspond to compact Lie groups.

If gC is a semisimple Lie algebra over the complex numbers, then there are several non-isomorphic real Lie algebras whose complexification is gC, 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.

For example, the complex special linear algebra sl(2, C) has two real forms, the real special linear algebra, denoted sl(2, R), and the 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.

See also[edit]

Notes[edit]

  1. ^ Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN 978-0-387-97527-6  See page 207.

References[edit]