$z$ when no confusion can arise.

The set $QO(V)$ is a nonassociative algebra with the operations $\circ_n$, which satisfy $1\circ_n a=\delta_{n,-1}a$ for all $n$, and $a\circ_n 1=\delta_{n,-1}a$ for $n\geq -1$. In particular, $1$ behaves as a unit with respect to $\circ_{-1}$. A linear subspace $\cA\subset QO(V)$ containing $1$ which is closed under the circle products will be called a [*quantum operator algebra*]{} (QOA). Note that $\cA$ is closed under $\partial$ since $\partial a=a\circ_{-2}1$. Many formal algebraic notions are immediately clear: a homomorphism is just a linear map that sends $1$ to $1$ and preserves all circle products; a module over $\cA$ is a vector space $M$ equipped with a homomorphism $\cA\rightarrow
QO(M)$, etc. A subset $S=\{a_i|\ i\in I\}$ of $\cA$ is said to generate $\cA$ if every element $a\in\cA$ can be written as a linear combination of nonassociative words in the letters $a_i$, $\circ_n$, for $i\in I$ and $n\in\mathbb{Z}$. We say that $S$ [*strongly generates*]{} $\cA$ if every $a\in\cA$ can be written as a linear combination of words in the letters $a_i$, $\circ_n$ for $n<0$. Equivalently, $\cA$ is spanned by the collection $\{ :\partial^{k_1} a_{i_1}(z)\cdots \partial^{k_m} a_{i_m}(z):| ~i_1,\dots,i_m \in I,~ k_1,\dots,k_m \geq 0\}$.

We say that $a,b\in QO(V)$ [*quantum commute*]{} 