Integral bases for the universal enveloping algebras of map algebras

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, U(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.