Weyl Modules for Toroidal Lie Algebras

Abstract

In this paper, we study Weyl modules for a toroidal Lie algebra \(\mathcal {T}\) with arbitrary n variables. Using the work of Rao (Pac. J. Math. 171(2), 511–528 1995), we prove that the level one global Weyl modules of \(\mathcal {T}\) are isomorphic to suitable submodules of a Fock space representation of \(\mathcal {T}\) up to a twist. As an application, we compute the graded character of the level one local Weyl module of \(\mathcal {T}\), thereby generalising the work of Kodera (Lett. Math. Phys. 110(11) 3053–3080 2020).