Weyl modules and Weyl functors for hyper-map algebras

TL;DR

This paper studies the structure and properties of Weyl modules and functors for hyper-map algebras, extending representation theory of Lie algebras to more general algebraic settings.

Contribution

It introduces and analyzes Weyl modules and functors for hyper-map algebras, establishing their universal properties and conditions for finite-dimensionality.

Findings

Weyl modules satisfy certain universal properties.

Conditions for local Weyl modules to be finite-dimensional.

Conditions for global Weyl modules to be finitely generated.

Abstract

We investigate the representations of the hyperalgebras associated to the map algebras $\mathfrak g\otimes \mathcal A$, where $\mathfrak g$ is any finite-dimensional complex simple Lie algebra and $\mathcal A$ is any associative commutative unitary algebra with a multiplicatively closed basis. We consider the natural definition of the local and global Weyl modules, and the Weyl functor for these algebras. Under certain conditions, we prove that these modules satisfy certain universal properties, and we also give conditions for the local or global Weyl modules to be finite-dimensional or finitely generated, respectively.