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Considering a Poisson algebra as a nonassociative algebra satisfying the Markl-Remm identity, we study deformations of Poisson algebras as deformations of this nonassociative algebra. We give a natural interpretation of deformations which preserve the underlying associative structure and of deformations which preserve the underlying Lie algebra and we compare the associated cohomologies with the Poisson cohomology parametrizing the general deformations of Poisson algebras.

We study the class of matrix controlled systems associated to graded filiform nilpotent Lie algebras. This generalizes the non- linear system corresponding to the control of the trails pulled by car.

In [8] we studied Koszulity of a family ${t𝒜\mathrm{𝑠𝑠}}_d^n$ of operads depending on a natural number $n\in \mathbb{N}$ and on the degree $d\in \mathbb{Z}$ of the generating operation. While we proved that, for $n\le 7$, the operad ${t𝒜\mathrm{𝑠𝑠}}_d^n$ is Koszul if and only if $d$ is even, and while it follows from [4] that ${t𝒜\mathrm{𝑠𝑠}}_d^n$ is Koszul for $d$ even and arbitrary $n$, the (non)Koszulity of ${t𝒜\mathrm{𝑠𝑠}}_d^n$ for $d$ odd and $n\ge 8$ remains an open problem. In this note we describe some related numerical experiments, and formulate a conjecture suggested by the results of these computations.

Flag manifolds are in general not symmetric spaces. But they are provided with a structure of ${\mathbb{Z}}_2^k$-symmetric space. We describe the Riemannian metrics adapted to this structure and some properties of reducibility. The conditions for a metric adapted to the ${\mathbb{Z}}_2^2$-symmetric structure to be naturally reductive are detailed for the flag manifold $SO\left(5\right)/SO\left(2\right)\times SO\left(2\right)\times SO\left(1\right)$.