How Can I Determine if Two Lie Algebras are Isomorphic?

[topsquark]

I now know of two 3D Lie algebras:

\(\displaystyle A_1\) with brackets \(\displaystyle \left [ T^0, ~T^{\pm} \right ] = \pm 2 T^{\pm} \), \(\displaystyle \left [ T^+, ~ T^- \right ] = T^0\)

and one with brackets:

\(\displaystyle \left [ T^0, ~T^{\pm} \right ] = T^{\mp}\) and \(\displaystyle \left [ T^+,~ T^- \right ] = T^0\)

How can I tell if these two are representations of the same thing? (Up to an isomorphism, anyway.) My thought is to show that the adjoint representation generators are similar. Using matrices that is to say a matrix S exists such that \(\displaystyle T_a' = ST^aS^{-1}\). Is this an acceptable way to show the existence of an isomorphism?

-Dan

 

[topsquark]

I don't think the algebra I found is isomorphic to A1. A1 has a 2 x 2 matrix representation but the other does not.

The similarity method seems to be out, so I was barking up the wrong tree. But even though I answered my own question here my problem still remains in general: How do I show if two Lie Algebras are the same? Is there a way to decompose a Lie algebra in a similar way to finding irreducible representations of a group?

-Dan