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Gröbner bases for operads. (English) Zbl 1208.18007
The purpose of the article under review is to set up a theory of Gröbner bases for symmetric operads. The authors notably succeed in defining a version of Burchberger’s algorithm for the construction of Gröbner bases in operadic ideals. The operadic Gröbner bases reduce to Hoffbeck’s notion of Poincaré-Birkhoff-Witt basis in the special case of ideals generated by quadratic operadic polynomials. According to E. Hoffbeck [Manuscr. Math. 131, No. 1–2, 87–110 (2010; Zbl 1207.18009)], an operad equipped with a Poincaré-Birkhoff-Witt basis is Koszul in the sense of V. Ginzburg and M. Kapranov [Duke Math. J. 76, No. 1, 203–272 (1994; Zbl 0855.18006)]. Thus, the algorithm defined in the article under review gives an effective process for the construction of Poincaré-Birkhoff-Witt basis of operads and, as a byproduct, gives an effective approach to check that an operad is Koszul. These observations, which parallel usual results on the classical homology theory of associative algebras, give authors’ main motivation for the definition of Gröbner bases in the context of operads.
In algebra, the definition of Gröbner bases relies on the existence of good ordering on monomials. For the extension to operads, we face the difficulty that operadic monomials have an intricate treewise structure, much more difficult to handle than the linear ordering of associative monomials, and we have to keep track of additional symmetric group actions in compositions. The authors work out these difficulties by extending ideas of E. Hoffbeck, which they put in a general conceptual framework. Notably, they regard the pointed shuffle composites of [loc. cit.] as the building blocks of the composition structure of a new category of operads, coined shuffle operads by them, which retain enough of the structure of symmetric operads for applications. More specifically, the authors observe that the forgetful functor from symmetric operads to shuffle operads preserve free objects, as well as presentations by generators and relations. Therefore, the category of shuffle operads gives a good effective device for the study of symmetric operads themselves.

MSC:
 18D50 Operads (MSC2010) 16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) 20B30 Symmetric groups
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