# Mobius: Representations of SU(2,1)=U(1,1)

I'm studying the representations of SU(2,1) [or U(1,1)], and since they are non-compact, their representations are necessarily infinite dimensional.

I have a couple questions.

In the literature, they say the algebra satisfied by the three generators, $T_1, T_2, T_3$ is

$$[T_1,\,T_2]=-i T_3$$
$$[T_2,\,T_3]=i T_1$$
$$[T_3,\,T_1]=i T_2$$​

Is this correct?

And, secondly, given that the group is non-compact, I should expect the generators to be represented by infinite-dimensional matrices only. But, in the literature, I found the following:

$$T_1 = \frac{1}{2}\begin{pmatrix}0&1\\-1&0\end{pmatrix}\qquad T_2=\frac{1}{2}\begin{pmatrix}0&-i\\-i&0\end{pmatrix}\qquad T_3=\frac{1}{2}\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$​

and these appear to satisfy the algebra above. So why aren't they infinite-dimensional? What's the deal?

So, su(2,1) can't be u(1,1). I would expect to be (2+1)^2 - 1 = 8-dimensional, and u(1,1) to be (1+1)^2 = 4-dimensional. The algebra you listed is 3d. It looks like it may very well be su(1,1) = so(2,1) = sl(2,R), I think. I expect one would get to u(1,1) by adding a central charge (a generator that commutes with everything), at which point the commutation relations wouldn't change. I did not verify that that algebra is consistent with the definition of su(1,1), but it certainly appears to be at first glance.

To answer your question, non-compact groups have UNITARY representations that are necessarily infinite-dimensional. They can have finite-dimensional representations that are not unitary. What you listed was an example of such a representation. The way to see whether a representation of a group is unitary by means of looking at the representation of the algebra alone is to ask whether or not the representation is hermitian. In this case, T3 is indeed hermitian, but T1 and T2 are not. The hermitian representations of this would be infinite-dimensional; for example, the action of this algebra on functions/fields on 2+1d spacetime. (This algebra would act as the lorentz algebra on those functions.)