12,675 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
Communication tree problems
In this paper, we consider random communication
requirements and several cost
measures for a particular model of tree routing on a
complete network. First
we show that a random tree does not give any approximation.
Then give
approximation algorithms for the case for two random models
of requirements.Postprint (published version
On a theorem of Kontsevich
In two seminal papers M. Kontsevich introduced graph homology as a tool to
compute the homology of three infinite dimensional Lie algebras, associated to
the three operads `commutative,' `associative' and `Lie.' We generalize his
theorem to all cyclic operads, in the process giving a more careful treatment
of the construction than in Kontsevich's original papers. We also give a more
explicit treatment of the isomorphisms of graph homologies with the homology of
moduli space and Out(F_r) outlined by Kontsevich. In [`Infinitesimal operations
on chain complexes of graphs', Mathematische Annalen, 327 (2003) 545-573] we
defined a Lie bracket and cobracket on the commutative graph complex, which was
extended in [James Conant, `Fusion and fission in graph complexes', Pac. J. 209
(2003), 219-230] to the case of all cyclic operads. These operations form a Lie
bi-algebra on a natural subcomplex. We show that in the associative and Lie
cases the subcomplex on which the bi-algebra structure exists carries all of
the homology, and we explain why the subcomplex in the commutative case does
not.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-42.abs.htm
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