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.

我们研究具有正特征的多电流和多环超代数的有限维表示类别，即与多电流代数\（\ mathfrak {g} \ otimes \ mathbb {C} [t_ {1}，\ ldots， t_ {n}] \）和多环代数\（\ mathfrak {g} \ otimes \ mathbb {C} [t_ {1} ^ {\ pm 1}，\ ldots，t_ {n} ^ {\ pm 1 }] \），其中\（\ mathfrak {g} \）是任何有限维复杂的简单Lie代数。主要结果是构造了通用有限维最高权重模块，并对每个类别中的不可约模块进行了分类。在特征零设置中，我们还提供了这些模块之间的关系。