Exploring $\mathcal{W}_{\infty}$ in the quadratic basis

We study the operator product expansions in the chiral algebra $\mathcal{W}_{\infty}$, first using the associativity conditions in the basis of primary generating fields and second using a different basis coming from the free field representation in which the OPE takes a simpler quadratic form. The results in the quadratic basis can be compactly written using certain bilocal combinations of the generating fields and we conjecture a closed-form formula for the complete OPE in this basis. Next we show that the commutation relations as well as correlation functions can be easily computed using properties of these bilocal fields. In the last part of this paper we verify the consistency with results derived previously by studying minimal models of $\mathcal{W}_{\infty}$ and comparing them to known reductions of $\mathcal{W}_{\infty}$ to $\mathcal{W}_N$. The results we obtain illustrate nicely the role of triality symmetry in the representation theory of $\mathcal{W}_{\infty}$.