A note on the affine vertex algebra associated to gl(1|1) at the critical level and its generalizations

Sa？etak In this note we present an explicit realization of the affine vertex algebra V^cri(gl(1|1)) inside of the tensor product F ？ M where F is a fermionic verex algebra and M is a commutative vertex algebra. This immediately gives an alternative description of the center of V^cri(gl(1|1)) as a subalgebra M_0 of M. We reconstruct the Molev-Mukhin formula for the Hilbert-Poincare series of the center of V^cri(gl(1|1)). Moreover, we construct a family of irreducible Vcri(gl(1|1))-modules realized on F and parameterized by χ+, χ- ∈ C((z)). We propose a generalization of V^cri(gl(1|1)) as a critical level version of the super W_{1+∞} vertex algebra