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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 Kähler...
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