Given a Lie algebra \(\mathfrak g\) over a field \(\mathbb K\), denote its universal enveloping algebra by \(U(\mathfrak g)\). An integral form of \(U(\mathfrak g)\) is a subring \(U_{\mathbb Z}(\mathfrak g)\) of \(U(\mathfrak g)\) such that \(\mathbb K \otimes_{\mathbb Z} U_{\mathbb Z} (\mathfrak g) \cong U(\mathfrak g)\). Several classes of Lie algebras are known to admit integral forms for their universal enveloping algebras, in particular, certain map algebras.
A map algebra is a Lie algebra constructed in the following way. Let \(\mathbb K\) be a field, \(\mathfrak g\) be a Lie algebra over \(\mathbb K\), and \(A\) be an associative, commutative, unital \(\mathbb K\)-algebra. Endow the \(\mathbb K\)-vector space \(\mathfrak g \otimes_{\mathbb K} A\) with the unique Lie bracket that extends: \[ [x \otimes a, y \otimes b] = [x, y] \otimes ab \quad \textup{ for all } x, y \in \mathfrak g, \ a, b \in A. \] Lie algebras obtained as above are known as map algebras and denoted by \(\mathfrak g_A\).
When the field \(\mathbb K\) is algebraically closed, has characteristic zero, the Lie algebra \(\mathfrak g\) is finite dimensional, simple, and the algebra \(A\) is finitely generated, then \(U(\mathfrak g_A)\) admits an integral form. In these cases, for each field \(\mathbb F\) one defines the \(\mathbb F\)-hyperalgebra of \(\mathfrak g_A\) to be \(U_{\mathbb F} (\mathfrak g_A) := \mathbb F \otimes_{\mathbb Z} U_{\mathbb Z} (\mathfrak g_A)\). In this paper, the authors study certain classes of modules for these hyperalgebras, which are also called hyper-map algebras.
The main class of modules studied in this paper is the one consisting of global Weyl modules (see Definition 2.8). The authors obtain generators and relations for these modules in Proposition 2.10, prove that they satisfy certain universal properties in Theorem 2.13, and compute their endomorphisms in Section 2.7. Amongst other things, the authors also define Weyl functors in Definition 2.11 and prove that they preserve finite-dimensionality in Theorem 2.14.