In Witten's article, 2+1-dimensional gravity as an exactly soluble system, I faced gauge algebras. Witten claims that for any scalar \(\lambda\), we have \(j_i\)s and \(P_i\)s related to each other as follows:

\([j_a ,j_b]=ε_{abc} j_c\)

\([j_a ,P_b]=ε_{abc} P_c\)

\([P_a ,P_b]=\lambdaε_{abc} j_c\)

He also claims, that in and only in $ISO(d-1,1)$ and $d=3$, do we have the following unique relations:

\([j_a ,j_b]=ε_{abc} j_c\)

\([P_a,P_b]=0, [j_a,P_b]=ε_{abc} P_c\)

How can I understand this? Any references?

This post imported from StackExchange Physics at 2015-03-29 12:41 (UTC), posted by SE-user Ali rezaie