Manin products, Koszul duality, Loday algebras and Deligne conjecture

Abstract

Abstract. In this article we give a conceptual definition of Manin products in any category endowed with two coherent monoidal products. This construction can be applied to associative algebras, non-symmetric operads, operads, colored operads, and properads presented by generators and relations. These two products, called black and white, are dual to each other under Koszul duality functor. We study their properties and compute several examples of black and white products for operads. These products allow us to define natural operations on the chain complex defining cohomology theories. With these operations, we are able to prove that Deligne's conjecture holds for a general class of operads and is not specific to the case of associative algebras. Finally, we prove generalized versions of a few conjectures raised by M. Aguiar and J.-L. Loday related to the Koszul property of operads defined by black products. These operads provide infinitely many examples for this generalized Deligne's conjecture.

Pre-jordan Algebras

“…All of them are called Loday algebras ( [49], or ABQR operad algebras in [23], [24]). These algebras have a common property of "splitting associativity", that is, expressing the multiplication of an associative algebra as the sum of a string of binary operations ( [41]).…”

Some results on L-dendriform algebras

“…All of them are called Loday algebras [16]. These algebras have a common property of ''splitting associativity'', that is, expressing the multiplication of an associative algebra as the sum of a string of binary operations [17].…”

“…Furthermore, it is reasonable to consider interpreting L-dendriform algebras in terms of the Manin black product of symmetric operads [16].…”

STANDORT Info

“…2-monoidal categories. The situation of a category with two compatible monoidal structures has been studied, although not with our examples in mind [AM10,Val08,JS93], and they are known as 2-monoidal categories. As always with higher categorical concepts, there is much leeway in the definitions as to what level of strictness one wants to require; Aguiar and Mahajan's definition of a 2-monoidal category [AM10, Section 6] is the laxest in the literature and fits our application, although in our case more strictness assumptions could be made.…”