Abstract
Unifying various generalizations of Reynolds operators, the relative cocycle weighted Reynolds operators are studied. We give a characterization of relative cocycle weighted Reynolds operators in the context of pre-Lie algebras. We construct an explicit graded Lie algebra whose Maurer-Cartan elements are given by a relative cocycle weighted Reynolds operator, which makes it possible to construct a cohomology for relative cocycle weighted Reynolds operators. This cohomology can be seen as the cohomology of a certain pre-Lie algebra with coefficients in a suitable representation. Finally, we introduce the notion of NS-pre-Lie algebras and show NS-pre-Lie algebras induce pre-Lie algebras and L-dendriform algebras.
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