36 research outputs found
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
Anick-type resolutions and consecutive pattern avoidance
For permutations avoiding consecutive patterns from a given set, we present a
combinatorial formula for the multiplicative inverse of the corresponding
exponential generating function. The formula comes from homological algebra
considerations in the same sense as the corresponding inversion formula for
avoiding word patterns comes from the well known Anick's resolution.Comment: 16 pages. Preliminary version, comments are welcom
Free resolutions via Gr\"obner bases
For associative algebras in many different categories, it is possible to
develop the machinery of Gr\"obner bases. A Gr\"obner basis of defining
relations for an algebra of such a category provides a "monomial replacement"
of this algebra. The main goal of this article is to demonstrate how this
machinery can be used for the purposes of homological algebra. More precisely,
our approach goes in three steps. First, we define a combinatorial resolution
for the monomial replacement of an object. Second, we extract from those
resolutions explicit representatives for homological classes. Finally, we
explain how to "deform" the differential to handle the general case. For
associative algebras, we recover a well known construction due to Anick. The
other case we discuss in detail is that of operads, where we discover
resolutions that haven't been known previously. We present various
applications, including a proofs 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: 34 pages, 4 figures. v2: added references to the work of
Drummond-Cole and Vallette. v3: added an explicit description of homology
classes in the monomial case and more examples, re-structured the exposition
to achieve more clarity. v4: changed the presentation of the main
construction to make it clearer, added another example (a computation of the
bar homology of Rota--Baxter algebras
Using homological duality in consecutive pattern avoidance
Using the approach suggested in [arXiv:1002.2761] we present below a
sufficient condition guaranteeing that two collections of patterns of
permutations have the same exponential generating functions for the number of
permutations avoiding elements of these collections as consecutive patterns. In
short, the coincidence of the latter generating functions is guaranteed by a
length-preserving bijection of patterns in these collections which is identical
on the overlappings of pairs of patterns where the overlappings are considered
as unordered sets. Our proof is based on a direct algorithm for the computation
of the inverse generating functions. As an application we present a large class
of patterns where this algorithm is fast and, in particular, allows to obtain a
linear ordinary differential equation with polynomial coefficients satisfied by
the inverse generating function.Comment: 12 pages, 1 figur