# Quadratic invariant in Chern-Simons Th. w/ ISO(2,1) group.

1. Apr 27, 2010

### zwicky

Hi dudes.

I'm studying the paper of Witten: 2+1 Dimensional gravity as an exactly soluble system.

Before eq (2.8) the author justifies as a way to find the inner product the fact that in this theory we have the casimir $$\epsilon_{abc}P^aJ^b$$. Then he introduce the invariant quadratic form
$$<J_a,P_b>=\delta_{ab}, <J_a,J_b>=0, <P_a,P_b>=0.$$
I', trying to realize about this step. Any comment would be very helpful. I looked at many books in group theory. I was able to construct the Killing metric but only with a cosmological constant, in the case of the Poincare group, it seems like the Killing metric is degenerate.

Thank you very much in advance.