958 research outputs found
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
Formality theorems for Hochschild complexes and their applications
We give a popular introduction to formality theorems for Hochschild complexes
and their applications. We review some of the recent results and prove that the
truncated Hochschild cochain complex of a polynomial algebra is non-formal.Comment: Submitted to proceedings of Poisson 200
Homotopy morphisms between convolution homotopy Lie algebras
In previous works by the authors, a bifunctor was associated to any operadic
twisting morphism, taking a coalgebra over a cooperad and an algebra over an
operad, and giving back the space of (graded) linear maps between them endowed
with a homotopy Lie algebra structure. We build on this result by using a more
general notion of -morphism between (co)algebras over a (co)operad
associated to a twisting morphism, and show that this bifunctor can be extended
to take such -morphisms in either one of its two slots. We also provide
a counterexample proving that it cannot be coherently extended to accept
-morphisms in both slots simultaneously. We apply this theory to
rational models for mapping spaces.Comment: 37 pages; v2: minor typo corrections, updated bibliography, final
versio
Canonical Proof nets for Classical Logic
Proof nets provide abstract counterparts to sequent proofs modulo rule
permutations; the idea being that if two proofs have the same underlying
proof-net, they are in essence the same proof. Providing a convincing proof-net
counterpart to proofs in the classical sequent calculus is thus an important
step in understanding classical sequent calculus proofs. By convincing, we mean
that (a) there should be a canonical function from sequent proofs to proof
nets, (b) it should be possible to check the correctness of a net in polynomial
time, (c) every correct net should be obtainable from a sequent calculus proof,
and (d) there should be a cut-elimination procedure which preserves
correctness. Previous attempts to give proof-net-like objects for propositional
classical logic have failed at least one of the above conditions. In [23], the
author presented a calculus of proof nets (expansion nets) satisfying (a) and
(b); the paper defined a sequent calculus corresponding to expansion nets but
gave no explicit demonstration of (c). That sequent calculus, called LK\ast in
this paper, is a novel one-sided sequent calculus with both additively and
multiplicatively formulated disjunction rules. In this paper (a self-contained
extended version of [23]), we give a full proof of (c) for expansion nets with
respect to LK\ast, and in addition give a cut-elimination procedure internal to
expansion nets - this makes expansion nets the first notion of proof-net for
classical logic satisfying all four criteria.Comment: Accepted for publication in APAL (Special issue, Classical Logic and
Computation
Five interpretations of Faà di Bruno's formula
International audienceIn these lectures we present five interpretations of the Fa' di Bruno formula which computes the n-th derivative of the composition of two functions of one variable: in terms of groups, Lie algebras and Hopf algebras, in combinatorics and within operads
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