Quillen homology for operads via Gröbner bases

Abstract

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 Gröbner basis. Next, we define a combinatorial resolution for the "monomial replacement" of each shuffle operad (provided by the Gröbner bases theory). Finally, we explain how to "deform" the differential to handle every operad with a Gröbner 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.

41 pages, this paper supersedes our previous preprint arXiv:0912.4895. Final version, to appear in Documenta Math

References in corpus (8)

• Koszul duality of operads and homology of partition posets
• Sur les A-infini catégories
• Shuffle algebras, homology, and consecutive pattern avoidance
• Koszul duality for Operads
• (Non-)Koszulness of operads for n-ary algebras, galgalim and other curiosities
• Free resolutions via Gröbner bases
• Implementing Gröbner bases for operads
• Operads for n-ary algebras - calculations and conjectures

Cited by in corpus (16)

• Deformations and homotopy theory of relative Rota-Baxter Lie algebras
• Algebraic structures of $F$-manifolds via pre-Lie algebras
• Shuffle algebras, homology, and consecutive pattern avoidance
• On generating series of finitely presented operads
• Cacti and filtered distributive laws
• Minimal models for monomial algebras
• A survey on deformations, cohomologies and homotopies of relative Rota-Baxter Lie algebras
• Veronese powers of operads and pure homotopy algebras
• Associator dependent algebras and Koszul duality
• Quadratic Algebras arising from Hopf operads generated by a single element
• Tangent complexes and the Diamond Lemma
• Generalized cohomological field theories in the higher order formalism
• Homotopy invariants for ${\overline{\mathcal{M}}}_{0,n}$ via Koszul duality
• Cohomological invariants of algebraic operads, I
• Reconnectads
• Wilf classes of non-symmetric operads