24,670 research outputs found
Manin products, Koszul duality, Loday algebras and Deligne conjecture
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.Comment: Final version, a few references adde
Quillen homology for operads via Gr\"obner bases
The main goal of this paper is to present a way to compute Quillen homology
of operads. The key idea is to use the notion of a shuffle operad we introduced
earlier; this allows to compute, for a symmetric operad, the homology classes
and the shape of the differential in its minimal model, although does not give
an insight on the symmetric groups action on the homology. Our approach goes in
several steps. First, we regard our symmetric operad as a shuffle operad, which
allows to compute its Gr\"obner basis. Next, we define a combinatorial
resolution for the "monomial replacement" of each shuffle operad (provided by
the Gr\"obner bases theory). Finally, we explain how to "deform" the
differential to handle every operad with a Gr\"obner basis, and find explicit
representatives of Quillen homology classes for a large class of operads. We
also present various applications, including a new proof of Hoffbeck's PBW
criterion, a proof of Koszulness for a class of operads coming from commutative
algebras, and a homology computation for the operads of Batalin-Vilkovisky
algebras and of Rota-Baxter algebras.Comment: 41 pages, this paper supersedes our previous preprint
arXiv:0912.4895. Final version, to appear in Documenta Mat
- …