  We describe representations of certain superconformal algebras in the
semi-infinite Weil complex related to the loop algebra of a complex
finite-dimensional Lie algebra and in the semi-infinite cohomology. We show
that in the case where the Lie algebra is endowed with a non-degenerate
invariant symmetric bilinear form, the relative semi-infinite cohomology of the
loop algebra has a structure, which is analogous to the classical structure of
the de Rham cohomology in K\"ahler geometry.
