Finite-Dimensional Representations of Hyper Multicurrent and Multiloop Algebras

Abstract

We investigate the categories of finite-dimensional representations of multicurrent and multiloop hyperalgebras in positive characteristic, i.e., the hyperalgebras associated to the multicurrent algebras $\mathfrak {g}\otimes \mathbb {C}[t_{1},\ldots ,t_{n}]$ and to the multiloop algebras $\mathfrak {g}\otimes \mathbb {C}[t_{1}^{\pm 1},\ldots ,t_{n}^{\pm 1}]$ , where $\mathfrak {g}$ is any finite-dimensional complex simple Lie algebra. The main results are the construction of the universal finite-dimensional highest-weight modules and a classification of the irreducible modules in each category. In the characteristic zero setting we also provide a relationship between these modules.