Important: new dates 21 - 22 October 2021

Weakly associative algebras and nonassociative Poisson algebras

Weakly associative algebras and nonassociative Poisson algebras In a previous work, we have presented Poisson algebras as nonassociative algebras that we have called Poisson-admissible algebras: the classical Poisson algebra structure (a commutative associative multiplication and a Lie bracket linked by the Leibniz identity) is obtained by the polarization process of the nonassociative multiplication and the depolarization gives the inverse process. In the latest work, we are interested in a larger class of nonassociative algebras called weakly associative algebras and characterized by an identity on the associator of the multiplication A(X,Y,Z)+A(Y,Z,X)-A(Y,X,Z)=0 This class contains in particular the Lie algebras, the associative algebras, the Poisson-admissible algebras and it is the smallest class of quadratic algebras containing the Lie algebras and the associative algebras. The polarization/depolarization process associates weakly associative algebras to nonassociative Poisson algebras (a commutative algebra, a Lie bracket linked by the Leibniz identity) in a one-to-one correspondence. As an application we study the formal deformation of commutative associative algebras in the category of weakly associative algebras, which “enlarges” the notion of deformation quantization of Poisson algebras. We finally examine algebraic properties of weakly associative algebras.
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L∞ Extensions for the Poisson algebra of a symplectic manifold

L∞ Extensions for the Poisson algebra of a symplectic manifold Let (M, ω) be a non-compact symplectic manifold. The universal central extension of the Lie algebra C∞(M) can be realized as the quotient Ω1(M)/δΩ2(M), where δ is the differential of the canonical homology complex. We will show how the Lie algebra Ω1(M)/δΩ2(M) extends to an L∞-algebra structure on the canonical homology complex. The extension procedure is inspired by a similar approach for general multisyplectic manifolds developed by Christopher Rogers, but is combinatorially more intricate and relies on results from symplectic Hodge theory. (j.w. Bas Janssens and Cornelia Vizman)
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Families of algebraic structures

Families of algebraic structures Algebraic structures may come into families, where each operation at hand is replaced by a family of operations indexed by some parameter set, which often bears a semigroup structure. I will introduce Rota-Baxter families, then address other family structures (dendriform, duplicial, pre-Lie,...), and finally give a general account of family algebras over a finitely presented linear operad, this operad together with its presentation naturally defining an algebraic structure on the set of parameters. Based on recent joint works with Loïc Foissy, Xing Gao and Yuanyuan Zhang.
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Neighborhoods of singular leaves, first return, normal forms.

Neighborhoods of singular leaves, first return, normal forms. TBA
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The universal Lie infinity- algebroid of a Lie Rinehart algebra and application to symetries of a foliation

The universal Lie infinity- algebroid of a Lie Rinehart algebra and application to symetries of a foliation TBA
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Rankin-Cohen deformations of graded algebras

Rankin-Cohen deformations of graded algebras Rankin-Cohen brackets are a classical object in the study of modular forms; it is a family of bidifferential operators allowing to construct from two modular forms of respective weights k and l a new modular form of weight k + l + 2n, for any integer n> 0. An important property of Rankin-Cohen brackets is that they constitute a formal deformation of the algebra of modular forms. We will present in the talk a survey on algebraic methods to construct Rankin-Cohen deformations on any commutative graduated algebra, and their links with the transvectants of the classical invariant theory. Applications in number theory concern not only modular forms, but also quasi-modular forms or weak Jacobi forms, and are the subject of joint works with YungJu Choie, François Martin and Emmanuel Royer.
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Singular foliations and constrained dynamical systems.

Singular foliations and constrained dynamical systems. TBA
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The L-infty algebra associated to a Leibniz algebra

The L-infty algebra associated to a Leibniz algebra An introductory talk with a simple construction to be shown. Based upon joint work with Thomas Strobl.
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