The Galileo Algebra is discussed in, for example, the wikipedia article Representation theory of the Galilean group. In that article, we can see that, for example, $$ [E,P^i]=0 $$ which means that translations commute, as one would expect. My question is, does the relation above imply $[H,P^i]=0$?
I would say that the answer is "no", as in general $$ \dot P^i\sim [H,P^i]\neq 0 $$ but I'm not sure how to make sense out of $[E,P^i]=0$ if $[H,P^i]\neq 0$. Is $E\neq H$? or maybe $P$ is not the canonical momentum, but some other operator instead?