Global Weyl modules for thin Lie algebras are finite-dimensional

TL;DR

This paper extends the concept of Weyl modules to a class of thin, graded Lie algebras and proves their global Weyl modules are finite-dimensional, with applications to polynomial Hamiltonian vector fields.

Contribution

It introduces a new class of thin Lie algebras and establishes the finite-dimensionality of their global Weyl modules, expanding the understanding of module categories.

Findings

Global Weyl modules are finite-dimensional for thin Lie algebras.

Introduces stratifications of module categories for these Lie algebras.

Identifies strata categories within the module categories.

Abstract

The notion of Weyl modules, both local and global, goes back to Chari and Pressley in the case of affine Lie algebras, and has been extensively studied for various Lie algebras graded by root systems. We extend that definition to a certain class of Lie algebras graded by weight lattices and prove that if such a Lie algebra satisfies a natural "thinness" condition, then already the global Weyl modules are finite-dimensional. Our motivating example of a thin Lie algebra is the Lie algebra of polynomial Hamiltonian vector fields on the plane vanishing at the origin. We also introduce stratifications of categories of modules over such Lie algebras and identify the corresponding strata categories.