38,461 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
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
Coulomb branches for rank 2 gauge groups in 3d N=4 gauge theories
The Coulomb branch of 3-dimensional N=4 gauge theories is the space of bare
and dressed BPS monopole operators. We utilise the conformal dimension to
define a fan which, upon intersection with the weight lattice of a GNO-dual
group, gives rise to a collection of semi-groups. It turns out that the unique
Hilbert bases of these semi-groups are a sufficient, finite set of monopole
operators which generate the entire chiral ring. Moreover, the knowledge of the
properties of the minimal generators is enough to compute the Hilbert series
explicitly. The techniques of this paper allow an efficient evaluation of the
Hilbert series for general rank gauge groups. As an application, we provide
various examples for all rank two gauge groups to demonstrate the novel
interpretation.Comment: v2: 98 pages, 30 figures, 34 tables, 1 appendix, matches JHEP versio
From an implicational system to its corresponding D-basis
Closure system is a fundamental concept appearing in several areas such as databases, formal concept analysis, artificial intelligence, etc. It is well-known that there exists a connection between a closure operator on a set and the lattice of its closed sets. Furthermore, the closure system can be replaced by a set of implications but this set has usually a lot of redundancy inducing non desired properties.
In the literature, there is a common interest in the search of the mini- mality of a set of implications because of the importance of bases. The well-known Duquenne-Guigues basis satisfies this minimality condition. However, several authors emphasize the relevance of the optimality in order to reduce the size of implications in the basis. In addition to this, some bases have been defined to improve the computation of closures relying on the directness property. The efficiency of computation with the direct basis is achieved due to the fact that the closure is computed in one traversal.
In this work, we focus on the D-basis, which is ordered-direct. An open problem is to obtain it from an arbitrary implicational system, so it is our aim in this paper. We introduce a method to compute the D-basis by means of minimal generators calculated using the Simplification Logic for implications.Universidad de Málaga. Campus de Excelencia Internacional AndalucĂa Tech. Supported by Grants TIN2011-28084 and TIN2014-59471-P of the Science and Innovation Ministry of Spain, which is co-financed by the European Social Fund
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