The Heisenberg algebra is essentially the canonical commutation relations (CCR) for bosons $[f,f^\dagger]=1$. $f$ is called an annihilation operator in physics ($f^\dagger$ creation operator). Normally, a physics textbook 'introduces' $f^k$ or $(f^\dagger)^k$ or any product of these operators. This product is associative and clearly it belongs to a different structure than to the algebra itself. Is it its universal enveloping algebra? If not what is it? I'm especially interested in the link to the CCR but it resembles the question of the role of a Casimir operator for, for instance, $su(1,1)$ or $su(2)$. Is it the same situation?
Please enlighten me and if you throw some references -- even better!