8 research outputs found
Operads with compatible CL-shellable partition posets admit a Poincar\'e-Birkhoff-Witt basis
In 2007, Vallette built a bridge across posets and operads by proving that an
operad is Koszul if and only if the associated partition posets are
Cohen-Macaulay. Both notions of being Koszul and being Cohen-Macaulay admit
different refinements: our goal here is to link two of these refinements. We
more precisely prove that any (basic-set) operad whose associated posets admit
isomorphism-compatible CL-shellings admits a Poincar\'e-Birkhoff-Witt basis.
Furthermore, we give counter-examples to the converse
Théorie de l'homotopie des opérades courbées et des algÚbres courbées
Curved algebras are algebras endowed with a predifferential, which is an endomorphism of degree â1 whose square is not necessarily zero. This makes the usual definition of quasi-isomorphism meaningless and therefore the homotopical study of curved algebras cannot follow the same path as differential graded algebras.In this article, we propose to study curved algebras by means of curved operads. We develop the theory of bar and cobar constructions adapted to this new notion as well as Koszul duality theory. To be able to provide meaningful definitions, we work in the context of objects which are filtered and complete and become differential graded after applying the associated graded functor. This setting brings its own difficulties but it nevertheless permits us to define a combinatorial model category structure that we can transfer to the category of curved operads and to the category of algebras over a curved operad using free-forgetful adjunctions. We address the case of curved associative algebras. We recover the notion of curved Aâ-algebras, and we show that the homotopy categories of curved associative algebras and of curved Aâ-algebras are Quillen equivalent
Operads with compatible CL-shellable partition posets admit a Poincaré-Birkhoff-Witt basis
In 2007, Vallette built a bridge across posets and operads by proving that an operad is Koszul if and only if the associated partition posets are Cohen-Macaulay. Both notions of being Koszul and being Cohen-Macaulay admit different refinements: our goal here is to link two of these refinements. We more precisely prove that any (basic-set) operad whose associated posets admit isomorphism-compatible CL-shellings admits a Poincaré-Birkhoff-Witt basis. Furthermore, we give counterexamples to the converse
Operads with compatible CL-shellable partition posets admit a Poincaré-Birkhoff-Witt basis
International audienceIn 2007, Vallette built a bridge across posets and operads by proving that an operad is Koszul if and only if the associated partition posets are Cohen-Macaulay. Both notions of being Koszul and being Cohen-Macaulay admit different refinements: our goal here is to link two of these refinements. We more precisely prove that any (basic-set) operad whose associated posets admit isomorphism-compatible CL-shellings admits a Poincaré-Birkhoff-Witt basis. Furthermore, we give counterexamples to the converse