Quillen homology for operads via gröbner bases

Vladimir Dotsenko, Anton Khoroshkin

Research output: Contribution to journalArticlepeer-review


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 fromcommutative algebras, and a homology computation for the operads of Batalin- Vilkovisky algebras and of Rota-Baxter algebras.

Original languageEnglish
Pages (from-to)707-747
Number of pages41
JournalDocumenta Mathematica
Issue number2013
StatePublished - 2013
Externally publishedYes

ASJC Scopus subject areas

  • General Mathematics


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