Killing form

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In mathematics, the Killing form, named for Wilhelm Killing (1847-1923), is a bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. In an example of Stigler's law of eponymy, the Killing form was actually invented by Élie Cartan, whereas the Cartan matrix is due to Wilhelm Killing.

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

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) = trace(ad(x)ad(y)),

with values in K, the Killing form on g.

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[edit] Properties

Note that B is symmetric and bilinear. The invariance property of Killing forms is the associativity property

B([x,y],z)=B(x,[y,z])

where [,] is the Lie bracket.

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 above properties imply that the Killing form is unique up to a multiplicative constant.

[edit] Matrix elements

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

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

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

Note that

\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

and so we can write

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

where the {c^{ij}}_{k} are the structure constants of the Lie algebra. Note that the Killing form is the simplest possible 2-tensor that can be formed from the structure constants.

Note that 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 correctly capturing the transformation properties of tensors.

[edit] Theorems

If B is nondegenerate, g is called semisimple. A Lie group whose Lie algebra is semisimple is called semisimple as well.

A Lie algebra g is said to be solvable if and only if B(x,x)= 0 for all x in the derived algebra [g,g].

If g and h are Lie algebras, then they are ideals in the direct sum g \oplus h and are thus orthogonal with respect to the Killing form. That is, B(x,y)=0 for all x in g and y in h. This implies that the Killing form of a semisimple algebra is determined by the Killing forms of its simple ideals.

When the Killing form is not degenerate, it 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.

[edit] Real forms

Any simple Lie algebra over the field of real numbers is not degenerate. This allows it to be brought into diagonal form with the appropriate choice of basis. That is, the Killing form can be written with only +1 or -1 on the diagonal, and all off-diagonal entries are zero.

If g is a simple Lie algebra over the complex numbers, then one can associate to it several different distinct Lie algebras over the reals. These are called the real forms of g.

There is one unique (up to isomorphism) real form for which the diagonal entries of the Killing form are all negative. This is sometimes called the compact real form of g. Here, compactness refers to the manifold of the corresponding Lie group.

For example, sl(2,C) has two real forms, sl(2,R) and su(2). The manifold of the Lie group SU(2) is the 3-sphere and is thus compact. Note that although the generators of su(2) are usually written with complex numbers, the algebra is still considered to be real because the commutation relations can be written so that no complex numbers appear in them. That is, su(2) can be written as an algebra over the real numbers, and the structure constants are real numbers.

[edit] References

  • William Fulton and Joe Harris, Representation Theory, A First Course, (1991) Springer-Verlag,New York. ISBN 0-387-97495-4
  • Jurgen Fuchs, Affine Lie Algebras and Quantum Groups, (1992) Cambridge University Press. ISBN 0-521-48412-X
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