Searcharxiv⌕ Search

arXiv subjects

Alexandre Quesney

Publications and source records attributed to Alexandre Quesney.

10 recordsLinked to original sources
arXiv2023

What is the Magnus Expansion?

The Magnus expansion, introduced by Wilhelm Magnus in 1954, is an infinite Lie series employed to express solutions for first-order homogeneous linear differential equations involving a linear operator. Since its discovery it has evolved into a pivotal tool used across diverse disciplines, including physics, chemistry, and engineering. Over the past 25 years, the Magnus expansion has undergone significant mathematical developments which revealed an intricate interplay between algebra, combinatorics, and geometry. By emphasizing a modern perspectives based on the use of pre- and post-Lie algebras, we discuss the Magnus expansion from the viewpoint of the notion of crossed morphism.

math-ph↗
arXiv2023

Balanced infinitesimal bialgebras, double Poisson gebras and pre-Calabi-Yau algebras

We consider the properad that governs the balanced infinitesimal bialgebras equipped with a coproduct of degree $1-d$. This properad naturally encodes a part of the structure of the pre-Calabi-Yau algebras of degree $d$. We compute the cobar construction of its Koszul dual coproperad and show that its gebras lie between the homotopy double Poisson gebras and the pre-Calabi-Yau algebras. Finally, we show that, if one is willing to consider their curved version, the two resulting notions of curved homotopy balanced infinitesimal bialgebra and curved homotopy double Poisson gebra are equivalent. A relationship with the homotopy odd Lie bialgebras is also discussed.

math.QA↗
arXiv2019

The quantum trace as a quantum non-abelianization map

We prove that the balanced Chekhov-Fock algebra of a punctured triangulated surface is isomorphic to a skein algebra which is a deformation of the algebra of regular functions of some abelian character variety. We first deduce from this observation a classification of the irreducible representations of the balanced Chekhov-Fock algebra at odd roots of unity, which generalizes to open surfaces the classification of Bonahon, Liu and Wong. We re-interpret Bonahon and Wong's quantum trace map as a non-commutative deformation of some regular morphism between this abelian character variety and the SL2-character variety. This algebraic morphism shares many resemblance with the non-abelianization map of Gaiotto, Moore, Hollands and Neitzke. When the punctured surface is closed, we prove that this algebraic non-abelianization map induces a birational morphism between a smooth torus and the relative SL2 character variety.

math.GT↗
arXiv2019

Classical shadows of stated skein representations at roots of unity

We extend some results of Bonahon, Bullock, Turaev and Wong concerning the skein algebras of closed surfaces to L^e's stated skein algebra associated to open surfaces. We prove that the stated skein algebra with deforming parameter +1 embeds canonically into the centers of the stated skein algebras whose deforming parameter is an odd root unity. We also construct an isomorphism between the stated skein algebra at +1 and the algebra of regular function of a generalization of the SL2-character variety of the surface. As a result, we associate to each isomorphism class of irreducible or local representations of the stated skein algebra, an invariant which is a point in the character variety.

math.GT↗
arXiv2016

On the deformation complex of homotopy affine actions

An affine action of an associative algebra $A$ on a vector space $V$ is an algebra morphism $A \to V \rtimes {\rm End}(V)$, where $V$ is a vector space and $V \rtimes {\rm End}(V)$ is the algebra of affine transformations of $V$. The one dimensional version of the Swiss-Cheese operad, denoted ${\mathrm{\bf{sc}}}_1$, is the operad that governs affine actions of associative algebras. This operad is Koszul and admits a minimal model denoted by $({\mathrm{\bf{sc}}}_1)_\infty$. Algebras over this minimal model are called Homotopy Affine Actions, they consist of an $A_\infty$-morphism $A \to V \rtimes {\rm End}(V)$, where $A$ is an $A_\infty$-algebra. In this paper we prove a relative version of Deligne's conjecture. In other words, we show that the deformation complex of a homotopy affine action has the structure of an algebra over an ${\rm SC}_2$ operad. That structure is naturally compatible with the ${\rm E}_2$ structure on the deformation complex of the $A_\infty$-algebra.

math.AT↗
arXiv2015

The relative lattice path operad

We construct a set-theoretic coloured operad that may be thought of as a combinatorial model for the Swiss Cheese operad. This is the relative (or Swiss Cheese) version of the lattice path operad constructed by Batanin and Berger. By adapting their condensation process we obtain a topological (resp. chain) operad that we show to be weakly equivalent to the topological (resp. chain) Swiss Cheese operad.

math.AT↗
arXiv2014

On the homology of the double cobar construction of a double suspension

The double cobar construction of a double suspension comes with a Connes-Moscovici structure, that is a homotopy G-algebra (or Gerstenhaber-Voronov algebra) structure together with a particular BV-operator up to a homotopy. We show that the homology of the double cobar construction of a double suspension is a free BV-algebra. In characteristic two, a similar result holds for the underlying $2$-restricted Gerstenhaber algebra. These facts rely on a formality theorem for the double cobar construction of a double suspension.

math.AT↗
arXiv2014

Homotopy BV-algebra structure on the double cobar construction

We show that the double cobar construction, $Ω^2 C_*(X)$, of a simplicial set $X$ is a homotopy BV-algebra if $X$ is a double suspension, or if $X$ is 2-reduced and the coefficient ring contains the ring of rational numbers $\mathbb{Q}$. Indeed, the Connes-Moscovici operator defines the desired homotopy BV-algebra structure on $Ω^2 C_*(X)$ when the antipode $S : ΩC_*(X) \to ΩC_*(X)$ is involutive. We proceed by defining a family of obstructions $O_n : \widetilde{C}_*(X) \to \widetilde{C}_*(X)^{\otimes n}$, $n\geq 2$ measuring the difference $S^2 - Id$. When $X$ is a suspension, the only obstruction remaining is $O_2 := E^{1,1} - τE^{1,1}$ where $E^{1,1}$ is the dual of the $\smile_1$-product. When $X$ is a double suspension the obstructions vanish.

math.AT↗