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Explicit formulae for cocycles of holomorphic vector fields with values in \(\lambda\) densities. (English) Zbl 1055.17010

The subject of this paper is cohomology of a Lie algebra of holomorphic vector fields on a punctured Riemann surface \(\Sigma_r\), with coefficients in a module of holomorphic \(\lambda\)-densities on \(\Sigma_r\).
Results about this cohomology were obtained by N. Kawazumi [Ann. Inst. Fourier 43, 655–712 (1993; Zbl 0782.57019)], but explicit cocycles were not known. Here they are provided for the second cohomology.
This is motivated by a search for generalizations of Krichever-Novikov algebras as corresponding abelian extensions, analogous to the generalization of Virasoro algebras by V. Ovsienko and C. Roger [Indag. Math. 9, 277–288 (1998; Zbl 0932.17029)].

MSC:

17B56 Cohomology of Lie (super)algebras
17B66 Lie algebras of vector fields and related (super) algebras
30F30 Differentials on Riemann surfaces
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