On the polynomial representations of current algebras

We obtain a sufficient condition for a representation of a Lie algebra of the form , where is a finite-dimensional simple Lie algebra over an algebraically closed field F of zero characteristic and Φ is an associative commutative algebra over F, to be polynomial. As a corollary we classify integrable graded irreducible modules over an extended loop algebra with finite-dimensional homogeneous spaces. The criteria is applied also to irreducible conformal modules over current algebras.