The $\mathcal{N}=1$ algebra $\mathcal{W}_\infty[\mu]$ and its truncations

The main objective of this work is to construct and classify the most general classical and quantum $\mathcal{N}=1$ $\mathcal{W}_\infty$-algebras generated by the same spins as the singlet algebra of $N$ fermions and $N$ bosons in the vector representation of $O(N)$ in the $N\to\infty$ limit. This type of algebras appears in a recent $\mathcal{N}=1$ version of the minimal model holography. Our analysis strongly suggests that there is a one parameter family $\mathcal{W}_\infty[\mu]$ of such algebras at every given central charge. Relying on this assumption, we identify various truncations of $\mathcal{W}_\infty[\mu]$ with, on the one hand, (orbifolds of) the Drinfel'd-Sokolov reductions of the Lie superalgebras $B(n,n)$, $B(n-1,n)$, $D(n,n)$ and $D(n+1,n)$, and, on the other hand, (orbifolds of) three $\mathcal{N}=1$ cosets. After a closer inspection we show that these cosets can be realized as a Drinfel'd-Sokolov reduction of $B(n,n)$, $D(n,n)$ and $D(n+1,n)$. We then discuss the implications of our findings for the quantum version of the $\mathcal{N}=1$ minimal model holography.