Semi-infinite cohomology and superconformal algebras

Poletaeva, Elena

Annales de l'Institut Fourier, Tome 51 (2001) no. 3, pp. 745-768.

Nous décrivons des représentations de certaines algèbres superconformes dans le complexe de Weil semi-infini de l'algèbre des lacets d'une algèbre de Lie complexe de dimension finie et dans la cohomologie semi-infinie. Nous démontrons que dans le cas où l'algèbre de Lie est munie d'une forme bilinéaire symétrique non dégénérée invariante, la cohomologie semi-infinie relative de l'algèbre des lacets admet une structure, qui est l'analogue de la structure classique de la cohomologie de de Rham des variétés kählériennes.

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 geometry.

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