Orbifolds and cosets of minimal $\mathcal{W}$-algebras

Let $\mathfrak{g}$ be a simple, finite-dimensional Lie (super)algebra equipped with an embedding of $\mathfrak{s}{\mathfrak{l}}_2$ inducing the minimal gradation on $\mathfrak{g}$. The corresponding minimal $\mathcal{W}$-algebra ${\mathcal{W}}^k(\mathfrak{g},e_{-\theta })$ introduced by Kac and Wakimoto has strong generators in weights $1,2,3\mathrm{/}2$, and all operator product expansions are known explicitly. The weight one subspace generates an affine vertex (super)algebra $V^{k^{\mathrm{\prime }}}({\mathfrak{g}}^{\mathrm{♮}})$ where ${\mathfrak{g}}^{\mathrm{♮}}\subset \mathfrak{g}$ denotes the centralizer of $\mathfrak{s}{\mathfrak{l}}_2$. Therefore $$\mathcal{W}$^k(\mathfrak{g}, e_{-\theta})$hasanactionofaconnectedLiegroup$G^{\natural}0$ with Lie algebra ${\mathfrak{g}}_0^{\mathrm{♮}}$, where ${\mathfrak{g}}_0^{\mathrm{♮}}$ denotes the even part of ${\mathfrak{g}}^{\mathrm{♮}}$. We show that for any reductive subgroup $G\subset G_0^{\mathrm{♮}}$, and for any reductive Lie algebra ${\mathfrak{g}}^{\mathrm{\prime }}\subset {\mathfrak{g}}^{\mathrm{♮}}$, the orbifold $$\mathcal{O}$^k = $\mathcal{W}$^k(\mathfrak{g}, e{-\theta})^{G}$andthecoset$$\mathcal{C}$^k = $\text{Com}$(V(\mathfrak{g}‘),$\mathcal{W}$^k(\mathfrak{g}, e_{-\theta}))$ are strongly finitely generated for generic values of $k$. Here $V({\mathfrak{g}}^{\mathrm{\prime }})$ denotes the affine vertex algebra associated to ${\mathfrak{g}}^{\mathrm{\prime }}$. We find explicit minimal strong generating sets for ${\mathcal{C}}^k$ when $\mathfrak{g}’ = \mathfrak{g}^{\natural}$and$\mathfrak{g}$iseither$\mathfrak{s} \mathfrak{l}n$,$\mathfrak{s}\mathfrak{p}{2n}$,$\mathfrak{s}\mathfrak{l}(2|n)$for$n\neq 2$,$\mathfrak{p}\mathfrak{s}\mathfrak{l}(2|2)$,or$\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|4)$. Finally, we conjecture some surprising coincidences among families of cosets ${\mathcal{C}}_k$ which are the simple quotients of ${\mathcal{C}}^k$, and we prove several cases of our conjecture.