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I understand the definition of a Killing Form $B$ as $B(X,Y)=Tr(ad(X)ad(Y))$

And when the Lie group is semi-simple the negative of the Killing Form can serve as a Riemannian metric.

{Wonder if thats why some people say that a semi-simple Lie Group has only one metric!}

I would like to know what is the difference between a "Killing Metric" and "Cartan-Killing metric" and the "Killing Form" ?

Or are they just different names for the same concept?

{Like the books by Knapp or Fulton and Harris have nothing called "Cartan-Killing metric"}

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3 Answers

up vote 8 down vote accepted

Let me add a few comments to the answers by Mariano and Theo.

There is a one-to-one correspondence between bi-invariant metrics (of any signature) in a Lie group and ad-invariant nondegenerate symmetric bilinear forms on its Lie algebra.

In a simple Lie algebra every nondegenerate symmetric bilinear form is proportional to the Killing form which you wrote down in your question, hence a simple Lie group has precisely one conformal class of bi-invariant metrics. If (and only if) the group is compact, are these metrics positive-definite. (Some people call this riemannian, reserving the word pseudo-riemannian (or sometimes also semi-riemannian) for indefinite signature metrics. Personally I prefer to use riemannian for general metrics.)

One can ask the question: which Lie groups admit bi-invariant metrics of any signature? which is the same thing as asking which Lie algebras admint ad-invariant non-degenerate symmetric bilinear forms. Such Lie algebras are called metric (or also sometimes quadratic, orthogonal,...) and although there is no classification except in small index (index 0 = positive-definite, index 1 = lorentzian, etc...) there is a structure theorem proved by Alberto Medina and Philippe Revoy in this paper (in French). Their theorem says that the class of such Lie algebras is generated by the simple and the one-dimensional Lie algebras under two operations: orthogonal direct sum and double extension, a construction explained in that paper. Double extension always results in indefinite signature, so if you are only interested in the positive-definite case, you get back to the well-known result that every positive-definite metric Lie algebra $\mathfrak{g}$ is isomorphic to the orthogonal direct sum of a compact semisimple Lie algebra and an abelian Lie algebra, or in other words, $$\mathfrak{g} \cong \mathfrak{s}_1 \oplus \cdots \oplus \mathfrak{s}_N \oplus \mathfrak{a}$$ where the $\mathfrak{s}_i$ are the simple factors and $\mathfrak{a}$ is abelian. Up to automorphisms, the most general positive-definite inner product on such a Lie algebra is given by choosing for each simple factor $\mathfrak{s}_i$ a positive multiple $\lambda_i > 0$ of the Killing form.

These Lie algebras are precisely the Lie algebras of compact Lie groups. Their metricity can also be understood as follows: take any positive-definite inner product on $\mathfrak{g}$ and averageng it over the adjoint representation.

So in summary, although there are metric Lie algebras which are not semisimple (or even reductive), their inner product is always an additional structure, unlike the Killing form which comes for free with the Lie algebra.

As for question concerning the difference between Killing form and Cartan(-Killing) metric it depends on who says this. In much of the Physics literature people refer to an inner product on a vector space as a "metric". But assuming that this is not the case, then the Killing form is a bilinear form on the Lie algebra, whereas the metric is a metric (in the sense of riemannian geometry) on the Lie group. If $G$ is a Lie group whose Lie algebra is semisimple, then the Killing form on its Lie algebra defines a bi-invariant metric on the Lie group, which I suppose you could call the Cartan-Killing metric.

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As for your first question in braces: a compact Lie group has bi-invariant metrics, and simple ones have unique bi-invariant metrics, so in this case that is the metric. (In semi-simple ones, one can scale the metrics of each factor indepedently, so there is no uniqueness)

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The "Killing Form" is due to Cartan. See e.g. the Wikipedia page, which begins "... In fact, the Killing form was actually invented by Élie Cartan, whereas the Cartan matrix is due to Wilhelm Killing."

Incidentally, a compact Lie group, as Mariano says above, has a bi-invariant metric. Any compact Lie group is a discrete quotient of a torus times a semisimple group. If there are torus components, then the Killing form is degenerate. If there are no torus parts, then the negative of the Killing form is a (positive-definite) metric.

On the other hand, in general the Killing form is indefinite on semisimple groups.

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