We introduce the concept of N-differential graded algebras ($N$-dga), and study the moduli space of deformations of the differential of a $N$-dga. We prove that it is controlled by what we call the $N$-Maurer-Cartan equation. We provide geometric examples such as the algebra of differential forms of depth $N$ on an affine manifold, and $N$-flat covariant derivatives. We also consider deformations of the differential of a $q$-differential graded algebra. We prove that it is controlled by a generalized Maurer-Cartan equation. We find explicit formulae for the coefficients involved in that equation. Deformations of the $3$-differential of $3$-differential graded algebras are controlled by the $(3,N)$ Maurer-Cartan equation. We find explicit formulae for the coefficients appearing in that equation, introduce new geometric examples of $N$-differential graded algebras, and use these results to study $N$-Lie algebroids. We study higher depth algebras, and work towards the construction of the concept of $A^N_ \infty$-algebras.[-]

Let $\mathfrak{h}$ be a finite dimensional real Leibniz algebra. Exactly as the linear dual space of a Lie algebra is a Poisson manifold with respect to the Kostant-Kirillov-Souriau (KKS) bracket, $\mathfrak{h}^*$ can be viewed as a generalized Poisson manifold. The corresponding bracket is roughly speaking the evaluation of the KKS bracket at $0$ in one variable. This (perhaps strange looking) bracket comes up naturally when quantizing $\mathfrak{h}^*$ in an analoguous way as one quantizes the dual of a Lie algebra. Namely, the product $X \vartriangleleft Y = exp(ad_X)(Y)$ can be lifted to cotangent level and gives than a symplectic micromorphism which can be quantized by Fourier integral operators. This is joint work with Benoit Dherin (2013). More recently, we developed with Charles Alexandre, Martin Bordemann and Salim Rivire a purely algebraic framework which gives the same star-product.[-]

Let $(H, R)$ be a finite dimensional quasitriangular Hopf algebra over a field $k$, and $_H\mathcal{M}$ the representation category of $H$. In this paper, we study the braided autoequivalences of the Drinfeld center $_H^H\mathcal{Y}\mathcal{D}$ trivializable on $_H\mathcal{M}$. We establish a group isomorphism between the group of those autoequivalences and the group of quantum commutative bi-Galois objects of the transmutation braided Hopf algebra $_RH$. We then apply this isomorphism to obtain a categorical interpretation of the exact sequence of the equivariant Brauer group $BM(k, H, R)$ established by Zhang. To this end, we have to develop the braided bi-Galois theory initiated by Schauenburg, which generalizes the Hopf bi-Galois theory over usual Hopf algebras to the one over braided Hopf algebras in a braided monoidal category.[-]