Uniqueness of bi-invariant metrics on Lie groups?

lie-groups riemannian-geometry

Solution 1:

The answer to your question comes from Schur's lemma, in the theory of group representations, applied to the adjoint representation of the Lie group.

The bi-invariant metric restricts to an Ad-invariant scalar product on the Lie algebra. Schur's lemma says that, in an irreducible representation, there is only one invariant symmetric bilinear form, up to a scalar multiple, (this is just one version of Schur's lemma, which has many other uses).

A Lie group is simple if and only if its adjoint representation is irreducible. Therefore, the answer to you question is : simple Lie group.

What if a compact Lie group is not simple. Well, then, it is a direct product of simple Lie groups and of a torus (afterwards, there can be a quotienting by a discrete subgroup). The cone of bi-invariant metrics is made up of positive linear combinations of the bi-invariant metrics of the factors in this direct product. -- Salem

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