In this talk we will consider a “differential counterpart” of the dendriform splitting procedure for operads. This problem has a very natural interpretation in the language of non-associative algebras. It is well-known that a (non-associative, in general) algebra equipped with a Rota-Baxter operator (a formalization of integration) gives rise to a system in a class of splitting algebras. The latter include dendriform (pre-associative), pre-Lie (left-symmetric), pre-Poisson, Zinbiel (pre-commutative) algebras, etc. What happens if we replace a Rota-Baxter operator with a derivation? The answer is well known for associative commutative algebras: the resulting class of systems obtained in this way coincides with the variety Nov of Novikov algebras. We will show in general that for an arbitrary binary operad Var the variety of derived Var-algebras coincides with the Manin white product of operads Var and Nov. If we allow the initial multiplication(s) to leave in the language of a derived algebra then the same sort of description can be obtained just by replacement of Nov with GD!, the Koszul dual to the operad of Gelfand-Dorfman algebras. We will also discuss similar statements for the “integral” case of Rota-Baxter operators.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
