  The essential feature of a root-graded Lie algebra L is the existence of a
split semisimple subalgebra g with respect to which L is an integrable module
with weights in a possibly non-reduced root system S of the same rank as the
root system R of g. Examples include map algebras (maps from an affine scheme
to g, S = R), matrix algebras like sl_n(A) for a unital associative algebra A
(S = R = A_{n-1}), finite-dimensional isotropic central-simple Lie algebras (S
properly contains R in general), and some equivariant map algebras.
  In this paper we study the category of representations of a root-graded Lie
algebra L which are integrable as representations of g and whose weights are
bounded by some dominant weight of g. We link this category to the module
category of an associative algebra, whose structure we determine for map
algebras and sl_n(A). Our results unify previous work of Chari and her
collaborators on map algebras and of Seligman on isotropic Lie algebras.
