1,140 research outputs found
Koszul duality for PROPs
The notion of PROP models the operations with multiple inputs and multiple
outputs, acting on some algebraic structures like the bialgebras or the Lie
bialgebras.
We prove a Koszul duality theory for PROPs generalizing the one for
associative algebras and for operads.Comment: submitted to the C. R. Acad. Sci. Pari
Analytical study of electrostatic ion beam traps
The use of electrostatic ion beam traps require to set many potentials on the
electrodes (ten in our case), making the tuning much more difficult than with
quadrupole traps. In order to obtain the best trapping conditions, an
analytical formula giving the electrostatic potential inside the trap is
required. In this paper, we present a general method to calculate the
analytical expression of the electrostatic potential in any axisymmetric set of
electrodes. We use conformal mapping to simplify the geometry of the boundary.
The calculation is then performed in a space of simple geometry. We show that
this method, providing excellent accuracy, allows to obtain the potential on
the axis as an analytic function of the potentials applied to the electrodes,
thus leading to fast, accurate and efficient calculations. We conclude by
presenting stability maps depending on the potentials that enabled us to find
the good trapping conditions for oxygen 4+ at much higher energies than what
has been achieved until now.Comment: 9 page
Brown's moduli spaces of curves and the gravity operad
This paper is built on the following observation: the purity of the mixed
Hodge structure on the cohomology of Brown's moduli spaces is essentially
equivalent to the freeness of the dihedral operad underlying the gravity
operad. We prove these two facts by relying on both the geometric and the
algebraic aspects of the problem: the complete geometric description of the
cohomology of Brown's moduli spaces and the coradical filtration of cofree
cooperads. This gives a conceptual proof of an identity of Bergstr\"om-Brown
which expresses the Betti numbers of Brown's moduli spaces via the inversion of
a generating series. This also generalizes the Salvatore-Tauraso theorem on the
nonsymmetric Lie operad.Comment: 26 pages; corrected Figure
Deformation theory of representations of prop(erad)s
We study the deformation theory of morphisms of properads and props thereby
extending to a non-linear framework Quillen's deformation theory for
commutative rings. The associated chain complex is endowed with a Lie algebra
up to homotopy structure. Its Maurer-Cartan elements correspond to deformed
structures, which allows us to give a geometric interpretation of these
results.
To do so, we endow the category of prop(erad)s with a model category
structure. We provide a complete study of models for prop(erad)s. A new
effective method to make minimal models explicit, that extends Koszul duality
theory, is introduced and the associated notion is called homotopy Koszul.
As a corollary, we obtain the (co)homology theories of (al)gebras over a
prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex
is endowed with a canonical Lie algebra up to homotopy structure in general and
a Lie algebra structure only in the Koszul case. In particular, we explicit the
deformation complex of morphisms from the properad of associative bialgebras.
For any minimal model of this properad, the boundary map of this chain complex
is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this
paper provides a complete proof of the existence of a Lie algebra up to
homotopy structure on the Gerstenhaber-Schack bicomplex associated to the
deformations of associative bialgebras.Comment: Version 4 : Statement about the properad of (non-commutative)
Frobenius bialgebras fixed in Section 4. [82 pages
Pre-Lie deformation theory
In this paper, we develop the deformation theory controlled by pre-Lie
algebras; the main tool is a new integration theory for pre-Lie algebras. The
main field of application lies in homotopy algebra structures over a Koszul
operad; in this case, we provide a homotopical description of the associated
Deligne groupoid. This permits us to give a conceptual proof, with complete
formulae, of the Homotopy Transfer Theorem by means of gauge action. We provide
a clear explanation of this latter ubiquitous result: there are two gauge
elements whose action on the original structure restrict its inputs and
respectively its output to the homotopy equivalent space. This implies that a
homotopy algebra structure transfers uniformly to a trivial structure on its
underlying homology if and only if it is gauge trivial; this is the ultimate
generalization of the -lemma.Comment: Final version. Minor corrections. To appear in the Moscow
Mathematical Journa
De Rham cohomology and homotopy Frobenius manifolds
We endow the de Rham cohomology of any Poisson or Jacobi manifold with a
natural homotopy Frobenius manifold structure. This result relies on a minimal
model theorem for multicomplexes and a new kind of a Hodge degeneration
condition.Comment: 11 pages, v2: added some reference
The minimal model for the Batalin-Vilkovisky operad
The purpose of this paper is to explain and to generalize, in a homotopical
way, the result of Barannikov-Kontsevich and Manin which states that the
underlying homology groups of some Batalin-Vilkovisky algebras carry a
Frobenius manifold structure. To this extent, we first make the minimal model
for the operad encoding BV-algebras explicit. Then we prove a homotopy transfer
theorem for the associated notion of homotopy BV-algebra. The final result
provides an extension of the action of the homology of the
Deligne-Mumford-Knudsen moduli space of genus 0 curves on the homology of some
BV-algebras to an action via higher homotopical operations organized by the
cohomology of the open moduli space of genus zero curves. Applications in
Poisson geometry and Lie algebra cohomology and to the Mirror Symmetry
conjecture are given.Comment: New section added containing applications to Poisson geometry, Lie
algebra cohomology and to the Mirror Symmetry conjecture. [36 pages, 4
figures
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