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 the affinization of the Lie superalgebra $\mathfrak{gl}(1|1)$. 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 $\mathfrak{gl}(1|1)$. We identify the invariants with affine supersymmetric polynomials via a version of the Chevalley theorem.

24 pages, final version; contribution to Rodney Baxter volume, J.Phys. A

References in corpus (3)

• Quantum toroidal ${\mathfrak{gl}}_1$ algebra : plane partitions
• On higher order Sugawara operators
• Elementary invariants for centralizers of nilpotent matrices

Cited by in corpus (6)

• Integrals of motion from quantum toroidal algebras
• Completeness of Bethe ansatz for Gaudin models associated with gl(1|1)
• Center of affine ${\mathfrak{sl}}_{2|1}$ at the critical level
• Segal-Sugawara vectors for orthosymplectic Lie superalgebras
• Dualities of Gaudin models with irregular singularities for general linear Lie (super)algebras
• Exactly Solved Models and Beyond: a special issue in honour of R J Baxter's 75th birthday