Direct sum structure of the super Virasoro algebra and a Fermion algebra arising from the quantum toroidal \(\mathfrak {gl}_2\)

It is known that the q-deformed Virasoro algebra can be constructed from a certain representation of the quantum toroidal \(\mathfrak {gl}_1\) algebra. In this paper, we apply the same construction to the quantum toroidal algebra of type \(\mathfrak {gl}_2\) and study the properties of resulting generators \(W_i(z)\) (\(i=1,2\)). The algebra generated by \(W_i(z)\) can be regarded as a q-deformation of the direct sum \(\textsf{F} \oplus \textsf{SVir}\), where \(\textsf{F}\) denotes the free fermion algebra and \(\textsf{SVir}\) stands for the \(N=1\) super Virasoro algebra, also referred to as the \(N=1\) superconformal algebra or the Neveu–Schwarz–Ramond algebra. Moreover, the generators \(W_i(z)\) admit two screening currents, and we show that their degeneration limits coincide with the screening currents of \(\textsf{SVir}\). We also establish quadratic relations satisfied by \(W_i(z)\) and show that they generate a pair of commuting q-deformed Virasoro algebras, which degenerate into two nontrivial commuting Virasoro algebras included in \(\textsf{F} \oplus \textsf{SVir}\).