There is an embedding of affine vertex algebras $V^k(\mathfrak{gl}_n) \hookrightarrow V^k(\mathfrak{sl}_{n+1})$, and the coset $\mathcal{C}^k(n) = \text{Com}(V^k(\mathfrak{gl}_n), V^k(\mathfrak{sl}_{n+1}))$ a natural generalization parafermion algebra $\mathfrak{sl}_2$. It was called generalized parafermions by third author shown to arise as one-parameter quotient universal two-parameter $\math...
