arXiv subjects

Anne Pichereau

Publications and source records attributed to Anne Pichereau.

6 recordsLinked to original sources

arXiv2017

Deformations of Poisson algebras

We study $\mathbb Z_2$-graded Poisson structures defined on $\mathbb Z_2$-graded commutative polynomial algebras. In small dimensional cases, we exhibit classifications of such Poisson structures, obtain the associated Poisson $\mathbb Z_2$-graded cohomology and in some cases, deformations of these Poisson brackets and $P_\infty$-algebra structures. We highlight differences and analogies between this $\mathbb Z_2$-graded context and the non graded context, by studying for example the links between Poisson cohomology and singularities.

math.QA↗

arXiv2013

Calabi-Yau algebras viewed as deformations of Poisson algebras

We define a family of 3-Calabi-Yau algebras by potentials. For some of these algebras, we explicitly compute the Hochschild homology with the help of Poisson homology. The point is that the Poisson potential has non-isolated singularities.

math.RA↗

arXiv2009

$L_\infty$-interpretation of a classification of deformations of Poisson structures in dimension three

We give an $L_\infty$-interpretation of the classification, obtained in [AP2], of the formal deformations of a family of exact Poisson structures in dimension three. We indeed obtain again the explicit formulas for all the formal deformations of these Poisson structures, together with a classification in the generic case, by constructing a suitable quasi-isomorphism between two $L_\infty$-algebras, which are associated to these Poisson structures.

math.QA↗

arXiv2008

Formal deformations of Poisson structures in low dimensions

In this paper, we study formal deformations of Poisson structures, especially for three families of Poisson varieties in dimensions two and three. For these families of Poisson structures, using an explicit basis of the second Poisson cohomology space, we solve the deformation equations at each step and obtain a large family of formal deformations for each Poisson structure which we consider. With the help of an explicit formula, we show that this family contains, modulo equivalence, all possible formal deformations. We show moreover that, when the Poisson structure is generic, all members of the family are non-equivalent.

math.QA↗

arXiv2007

Double Poisson Cohomology of Path Algebras of Quivers

In this note, we give a description of the graded Lie algebra of double derivations of a path algebra as a graded version of the necklace Lie algebra equipped with the Kontsevich bracket. Furthermore, we formally introduce the notion of double Poisson-Lichnerowicz cohomology for double Poisson algebras, and give some elementary properties. We introduce the notion of a linear double Poisson tensor on a quiver and show that it induces the structure of a finite dimensional algebra on the vector spaces V_v generated by the loops in the vertex v. We show that the Hochschild cohomology of the associative algebra can be recovered from the double Poisson cohomology. Then, we use the description of the graded necklace Lie algebra to determine the low-dimensional double Poisson-Lichnerowicz cohomology groups for three types of (linear and non-linear) double Poisson brackets on the free algebra in two variables. This allows us to develop some useful techniques for the computation of the double Poisson-Lichnerowicz cohomology.

math.RA↗

arXiv2005

Poisson (co)homology and isolated singularities

To each polynomial $\v\in\F[x,y,z]$ is associated a Poisson structure on $\F^3$, a surface and a Poisson structure on this surface. When $\v$ is weight homogeneous with an isolated singularity, we determine the Poisson cohomology and homology of the two Poisson varieties obtained.

math.QA↗