Invariants of the vacuum module associated with the Lie superalgebra gl(1|1)

Abstract

We describe the algebra of invariants of the vacuum module associated with an affinization of the Lie superalgebra gl(1|1). We give a formula for its Hilbert-Poincaré series in a fermionic (cancellation-free) form which turns out to coincide with the generating function of the plane partitions over the (1, 1)-hook. Our arguments are based on a super version of the Beilinson-Drinfeld-Raïs-Tauvel theorem which we prove by producing an explicit basis of invariants of the symmetric algebra of polynomial currents associated with gl(1|1). We identify the invariants with affine supersymmetric polynomials via a version of the Chevalley theorem.

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

“…We recall the definition of universal affine vertex algebra V cri (gl(m|n)) following notations in [15].…”

“…But the case m = n is different. It was observed in [15] and [16] that then one needs to take central element K acting as identity on V cri (g).…”

“…In the recent paper [15], A. Molev and E. Mukhin determined the structure of Z(g) in the case g = gl(1|1), and presented conjectures on the structure of the center for g = gl(m|n). In the case g = gl(1|1), they constructed a family of central, Sugawara elements (see also [16]) and proved that these central elements generate Z(g) (see [15,Theorem 2.1]).…”