Given a finite-dimensional, simple Lie algebra g over C and A, a commutative, associative algebra with unity over C, we exhibit an integral form for the universal enveloping algebra of the map algebra, g ⊗ A, and an explicit Z-basis for this integral form. We also produce explicit commutation formulas in the universal enveloping algebra of sl2 ⊗ A that allow us to write certain elements in Poincaré-Birkhoff-Witt order.

Abstract. Let g be a finite dimensional complex simple classical Lie superalgebra and A be a commutative, associative algebra with unity over C. In this paper we define an integral form for the universal enveloping algebra of the map superalgebra g ⊗ A, and exhibit an explicit integral basis for this integral form.

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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 g⊗C[t 1 , . . . , t n ] and to the multiloop algebras g⊗C[t ±1 1 , . . . , t ±1 n ], where 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 irreducible modules in each category. In the characteristic zero setting we also provide a relationship between them.A.B. is partially supported by CNPq grant 462315/2014-2 and FAPESP grants 2015/22040-0 and 2014/09310-5. 1 REPRESENTATIONS OF HYPER MULTICURRENT AND MULTILOOP ALGEBRAS 2The goal of this paper is to establish basic results about the finite-dimensional representations of multicurrent and multiloop hyperalgebras, which have the form g ⊗ C[t 1 , . . . , t n ] and g ⊗ C[t ±1 1 , . . . , t ±1 n ] respectively, extending some of the known results in the case of current and loop algebras (g ⊗ C[t] and g ⊗ C[t ±1 ]). The approach for this is similar to [17,6], with the remark that the known characteristic zero methods as those compiled in [9] are not available for the hyperalgebra setting. We denote the multicurrent algebra in n variables by g[n] and the multiloop algebra by g n . Our main results are the construction of the universal finite-dimensional highest-weight modules in the categories of finite-dimensional

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