Wagemann, Friedrich Explicit formulae for cocycles of holomorphic vector fields with values in \(\lambda\) densities. (English) Zbl 1055.17010 J. Lie Theory 11, No. 1, 173-184 (2001). 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)]. Reviewer: Pasha Zusmanovich (Amsterdam) Cited in 1 ReviewCited in 4 Documents MSC: 17B56 Cohomology of Lie (super)algebras 17B66 Lie algebras of vector fields and related (super) algebras 30F30 Differentials on Riemann surfaces Keywords:Lie algebra of vector fields; densities; 2-cocycles; punctured Riemann surface; Krichever-Novikov algebras Citations:Zbl 0782.57019; Zbl 0932.17029 PDFBibTeX XMLCite \textit{F. Wagemann}, J. Lie Theory 11, No. 1, 173--184 (2001; Zbl 1055.17010) Full Text: arXiv EuDML