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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 a Z-form for the universal

enveloping algebra of the map algebra for g and an

explicit Z-basis for this Z-form. We also produce

explicit commutation formulas in the universal enveloping

algebra of the map algebra of sl2 that allow us to write certain

elements in Poincaré-Birkhoff-Witt order.

Finally we give some applications of these formulas to the

representation theory of the map algebras for sl2.

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