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# Documents  Makhlouf, Abdenacer | enregistrements trouvés : 12

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## The Weil algebra of a Hopf algebra Dubois-Violette, Michel | CIRM

Post-edited

Research talks;Algebra;Algebraic and Complex Geometry;Mathematical Physics

We give a summary of a joint work with Giovanni Landi (Trieste University) on a non commutative generalization of Henri Cartan's theory of operations, algebraic connections and Weil algebra.

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## Triality Elduque, Alberto | CIRM H

Post-edited

Research talks;Algebra

Duality in projective geometry is a well-known phenomenon in any dimension. On the other hand, geometric triality deals with points and spaces of two different kinds in a sevendimensional projective space. It goes back to Study (1913) and Cartan (1925), and was soon realizedthat this phenomenon is tightly related to the algebra of octonions, and the order 3 outer automorphisms of the spin group in dimension 8.
Tits observed, in 1959, the existence of two different types of geometric triality. One of them is related to the octonions, but the other one is better explained in terms of a class of nonunital composition algebras discovered by the physicist Okubo (1978) inside 3x3-matrices, and which has led to the definition of the so called symmetric composition algebras.
This talk will review the history, classification, and their connections with the phenomenon of triality, of the symmetric composition algebras.
Duality in projective geometry is a well-known phenomenon in any dimension. On the other hand, geometric triality deals with points and spaces of two different kinds in a sevendimensional projective space. It goes back to Study (1913) and Cartan (1925), and was soon realizedthat this phenomenon is tightly related to the algebra of octonions, and the order 3 outer automorphisms of the spin group in dimension 8.
Tits observed, in 1959, the ...

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## Gravitation: théorie et expérience.Troisième école de physique théoriqueJijel # 26 septembre - 3 octobre 2009 Bounames, Abdelhafid ; Makhlouf, Abdenacer ; Dubois-Violette, Michel | Hermann 2013

Congrès

- xi; 448 p.
ISBN 978-2-7056-8049-7

Travaux en cours , 0079

Localisation : Colloque 1er étage (JIJE)

relativité générale # gravité # gravité quantique # gravité quantique à boucles # théorie des cordes # cosmologie # onde gravitationnelle # algèbre # cohomologie # déformation formelle # variété différentiable # structure complexe # structure quaternionique # tenseur énergie-impulsion

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## Algebra, geometry and mathematical physics (AGMP)Mulhouse # october 2011 Makhlouf, Abdenacer ; Paal, Eugen ; Silvestrov, Sergei D. ; Stolin, Alexander | Springer 2014

Congrès

- xviii; 684 p.
ISBN 978-3-642-55360-8

Springer proceedings in mathematics & statistics , 0085

Localisation : Colloque 1er étage (MULH)

déformation # quantification # algèbre de Hom # structure algébrique # algèbre de Hopf # algèbre quantique # système intégrable # théorie des jets # théorie de Lie # algèbre de Lie non commutative # physique mathématique # algèbre ternaire

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## Deformations of $N$-differential graded algebras Díaz, Rafael | CIRM

Multi angle

Research talks;Algebra;Algebraic and Complex Geometry;Mathematical Physics

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. 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 ...

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## Deformation quantization of Leibniz algebras Wagemann, Friedrich | CIRM

Multi angle

Research talks;Algebra;Topology

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 $\mathfrak{h}$ 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 ...

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## Braided autoequivalences, quantum commutative Galois objects and the Brauer groups Zhang, Yinhuo | CIRM

Multi angle

Research talks;Algebra

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. 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 ...

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## Nakayama automorphism and quantum group actions on Artin-Schelter regular algebras Zhang, James | CIRM

Multi angle

Research talks;Algebra

The Nakayama automorphism of an Artin-Schelter regular algebra $A$ controls the class of quantum groups that act on the algebra $A$. Several applications are given.

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## Noncommutative localization in smooth Deformation quantization Bordemann, Martin | CIRM H

Multi angle

Research talks;Analysis and its Applications;Algebra;Geometry;Mathematical Physics

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## On the geometry of noncommutative deformations Boucetta, Mohamed | CIRM H

Multi angle

Research talks;Geometry

53D17

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## Finite quantum geometry, exceptional quantum geometry and fundamental particles Dubois-Violette, Michel | CIRM H

Multi angle

Research talks;Algebra;Algebraic and Complex Geometry;Lie Theory and Generalizations;Mathematical Physics

We show that the spectrum of fundamental particles of matter and their symmetries can be encoded in a finite quantum geometry equipped with a supplementary structure connected with the quark-lepton symmetry. The occurrence of the exceptional quantum geometry for the description of the standard model with 3 generations is suggested. We discuss the field theoretical aspect of this approach taking into account the theory of connections on the corresponding Jordan modules. We show that the spectrum of fundamental particles of matter and their symmetries can be encoded in a finite quantum geometry equipped with a supplementary structure connected with the quark-lepton symmetry. The occurrence of the exceptional quantum geometry for the description of the standard model with 3 generations is suggested. We discuss the field theoretical aspect of this approach taking into account the theory of connections on the ...

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## Closed $G_{2}$-structures Fino, Anna | CIRM H

Multi angle

Research talks;Algebra;Geometry;Lie Theory and Generalizations

I will review known examples of compact 7-manifolds admitting a closed $G_{2}$-structure. Moreover, I will discuss some results on the behaviour of the Laplacian $G_{2}$-flow starting from a closed $G_{2}$-structure whose induced metric satisfies suitable extra condition.

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