The Cohomology of relative cocycle weighted Reynolds operators and NS-pre-Lie algebras

Abstract: Unifying various generalizations of the important notions of Reynolds operators, the relative cocycle weighted Reynolds operators are studied. Here cocycle weighted means the weight of the operators is given by a 2-cocycle rather than by a scaler as in the classical case. We show that the operators and 2-cocycles uniquely determine each other. We further give a characterization of relative cocycle weighted Reynolds operators in the context of pre-Lie algebras. Using a method of Liu, we construct an explicit graded Lie algebra whose Maurer-Cartan elements are given by a relative cocycle weighted Reynolds operator. This allows us to construct the cohomology for a relative cocycle weighted Reynolds operator. This cohomology can also be seen as the cohomology of a certain pre-Lie algebra with coefficients in a suitable representation. Then we consider formal deformations of relative cocycle weighted Reynolds operators from cohomological points of view. Finally, we introduce the notation of NS-pre-Lie algebras and show NS-pre-Lie algebras naturally induce pre-Lie algebras and $L$-dendriform algebras.