I am taking a class on quantum mechanics and the following is stated: The space $\mathcal{F}$ of wavefunctions we will consider is a subspace of $L^2$ with good regularity properties (e.g. Schwartz space). The space is equipped with the standard scalar product $\langle \cdot , \cdot\rangle$. We will also consider operators acting on those wavefunctions, and associated to those operators is a commutator. Namely, if $A$, $B$ are two operators, then $[A,B]=AB-BA$.
My question is: This commutator looks a lot like a Lie bracket. Is it one? If so, is the space of operators a Lie algebra, and can we use Lie algebra techniques to study it? One idea that immediately comes to mind is to use representation theory to study those operators. I'm a mathematics student so I have little background in physics, sorry if this question sounds naive or obvious. In turn, physical intuition is probably lost on me as well.