These notes describe a family of some kind of extensions of the affine super Lie algebra \(\widehat{\mathfrak{gl}}(1|1)\), which can be realized algorithmically using the fusion rules of \({\mathfrak{gl}}(1|1)\) that give rise an infinite family of simple currents. Based on a previous work, the authors perform as an example these computations up to a certain order using a free field realization and study the \(W\)-algebras. In particular, they show that for certain families of parameters, these algebras contain a bosonic subalgebra and it is conjectured that this subalgebra equals the \(W^{(2)}_{N}\) algebra of Feigin and Semikhatov.
For the entire collection see [Zbl 1266.00027].