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Yael Fregier

Publications and source records attributed to Yael Fregier.

13 recordsLinked to original sources
arXiv2019

Homotopy theory of singular foliations

In this article we apply ideas from homotopy theory to the study of singular foliations. We verify that a technical lemma remains valid for left semi-model categories. When applied to the category of $L_\infty$-algebroids thanks to the work of Nuiten, this lemma enables to recover results very similar to those of Laurent-Gengoux and al. about existence and (up to homotopy) uniqueness of universal $L_\infty$-algebroids associated to a singular foliation. We conclude with some open questions.

math.AT↗
arXiv2019

Cohomologies of a Lie algebra with a derivation and applications

The main object of study of this paper is the notion of a LieDer pair, i.e. a Lie algebra with a derivation. We introduce the concept of a representation of a LieDer pair and study the corresponding cohomologies. We show that a LieDer pair is rigid if the second cohomology group is trivial, and a deformation of order n is extensible if its obstruction class, which is defined to be an element is the third cohomology group, is trivial. We classify central extensions of LieDer pairs using the second cohomology group with the coefficient in the trivial representation. For a pair of derivations, we define its obstruction class and show that it is extensible if and only if the obstruction class is trivial. Finally, we classify Lie2Der pairs using the third cohomology group of a LieDer pair.

math.RT↗
arXiv2016

Homotopy moment maps

Associated to any manifold equipped with a closed form of degree >1 is an `L-infinity algebra of observables' which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a map is a L-infinity morphism from the Lie algebra of the group into the observables which lifts the infinitesimal action. We establish the relationship between homotopy moment maps and equivariant de Rham cohomology, and analyze the obstruction theory for the existence of such maps. This allows us to easily and explicitly construct a large number of examples. These include results concerning group actions on loop spaces and moduli spaces of flat connections. Relationships are also established with previous work by others in classical field theory, algebroid theory, and dg geometry. Furthermore, we use our theory to geometrically construct various L-infinity algebras as higher central extensions of Lie algebras, in analogy with Kostant's quantization theory. In particular, the so-called `string Lie 2-algebra' arises this way.

math.DG↗
arXiv2015

Simultaneous deformations of algebras and morphisms via derived brackets

We present a method to construct explicitly L-infinity algebras governing simultaneous deformations of various kinds of algebraic structures and of their morphisms. It is an alternative to the heavy use of the operad machinery of the existing approaches. Our method relies on Voronov's derived bracket construction.

math.QA↗
arXiv2015

A cohomological framework for homotopy moment maps

Given a Lie group acting on a manifold $M$ preserving a closed $n+1$-form $ω$, the notion of homotopy moment map for this action was introduced in Callies-Fregier-Rogers-Zambon [6], in terms of $L_{\infty}$-algebra morphisms. In this note we describe homotopy moment maps as coboundaries of a certain complex. This description simplifies greatly computations, and we use it to study various properties of homotopy moment maps: their relation to equivariant cohomology, their obstruction theory, how they induce new ones on mapping spaces, and their equivalences. The results we obtain extend some of the results of [6].

math.DG↗
arXiv2015

Lower Central Series Ideal Quotients Over F_p and Z

Given a graded associative algebra $A$, its lower central series is defined by $L_1 = A$ and $L_{i+1} = [L_i, A]$. We consider successive quotients $N_i(A) = M_i(A) / M_{i+1}(A)$, where $M_i(A) = AL_i(A) A$. These quotients are direct sums of graded components. Our purpose is to describe the $\mathbb{Z}$-module structure of the components; i.e., their free and torsion parts. Following computer exploration using {\it MAGMA}, two main cases are studied. The first considers $A = A_n / (f_1,\dots, f_m)$, with $A_n$ the free algebra on $n$ generators $\{x_1, \ldots, x_n\}$ over a field of characteristic $p$. The relations $f_i$ are noncommutative polynomials in $x_j^{p^{n_j}},$ for some integers $n_j$. For primes $p > 2$, we prove that $p^{\sum n_j} \mid \text{dim}(N_i(A))$. Moreover, we determine polynomials dividing the Hilbert series of each $N_i(A)$. The second concerns $A = \mathbb{Z} \langle x_1, x_2, \rangle / (x_1^m, x_2^n)$. For $i = 2,3$, the bigraded structure of $N_i(A_2)$ is completely described.

math.RT↗
arXiv2014

Simultaneous deformations and Poisson geometry

We consider the problem of deforming simultaneously a pair of given structures. We show that such deformations are governed by an L-infinity algebra, which we construct explicitly. Our machinery is based on Th. Voronov's derived bracket construction. In this paper we consider only geometric applications, including deformations of coisotropic submanifolds in Poisson manifolds, of twisted Poisson structures, and of complex structures within generalized complex geometry. These applications can not be, to our knowledge, obtained by other methods such as operad theory.

math.QA↗
arXiv2013

Non abelian cohomology of extensions of Lie algebras as Deligne groupoid

In this note we show that the theory of non abelian extensions of a Lie algebra $\mathfrak{g}$ by a Lie algebra $\mathfrak{h}$ can be understood in terms of a differential graded Lie algebra $L$. More precisely we show that the non-abelian cohomology $H^2_{nab}(\mathfrak{g},\mathfrak{h})$ is $\mathcal{MC}(L)$, the $π_0$ of the Deligne groupoid of $L$.

math.RT↗
arXiv2010

On Hopf 2-algebras

Our main goal in this paper is to translate the diagram relating groups, Lie algebras and Hopf algebras to the corresponding 2-objects, i.e. to categorify it. This is done interpreting 2-objects as crossed modules and showing the compatibility of the standard functors linking groups, Lie algebras and Hopf algebras with the concept of a crossed module. One outcome is the construction of an enveloping algebra of the string Lie algebra of Baez-Crans, another is the clarification of the passage from crossed modules of Hopf algebras to Hopf 2-algebras.

math.GR↗
arXiv2009

On unitality conditions for Hom-associative algebras

In hom-associative structures, the associativity condition $(xy)z=x(yz)$ is twisted to $α(x)(yz) = (xy)α(z)$, with $α$ a map in the appropriate category. In the present paper, we consider two different unitality conditions for hom-associative algebras. The first one, existence of a unit in the classical sense, is stronger than the second one, which we call weak unitality. We show associativity conditions connected to the size of the image of the twisting map for unital hom-associative algebras. Also the problem of embedding arbitrary hom-associative algebras into unital or weakly unital ones is investigated. Finally, we show that weakly unital hom-associative algebras with bijective twisting map are twisted versions of associative algebras.

math.RA↗
arXiv2007

Decomposition of symmetric tensor fields in the presence of a flat contact projective structure

Let $M$ be an odd-dimensional Euclidean space endowed with a contact 1-form $α$. We investigate the space of symmetric contravariant tensor fields on $M$ as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up by those vector fields that preserve the contact structure. If we consider symmetric tensor fields with coefficients in tensor densities, the vertical cotangent lift of contact form $α$ is a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symmetric density valued tensor fields. This generalized Hamiltonian operator on the symbol space is invariant with respect to the action of the projective contact algebra $sp(2n+2)$. The preceding invariant operators lead to a decomposition of the symbol space (expect for some critical density weights), which generalizes a splitting proposed by V. Ovsienko.

math.DG↗
arXiv2004

Deformation of the Heisenberg Algebra inside $gl(3,\mathbb{K})$

We study non-trivial deformations of the natural imbedding of the Lie algebra $\fh_1$ of lower triangular matrices (the Heisenberg Lie algebra) into $gl(3,\mathbb{K})$, where $\mathbb{K}=\mathbb{R}$ or $|mathbb{C}$. Our first result is the calculation of the first cohomology space $H^1(\fh_1;gl(3,\mathbb{K}))$. We prove that there are no obstructions for integrability of infinitesemal deformation and, furthermore, give an explicit formula for the most general deformation.

math.RT↗