arXiv subjects

Didier Arnal

Publications and source records attributed to Didier Arnal.

14 recordsLinked to original sources

arXiv2026

$\mathfrak{sl}(2)$-cohomology on multidifferential operators space

In this paper, we consider the known question of the computation of the cohomology spaces for the natural action of the Lie algebra $\mathfrak{sl}(2)$ acting on the space $\Dlm$ of multidifferential operators on the line. We first show that these cohomology spaces are entirely characterized by a linear map $f$ between two explicit finite dimensional spaces, then we prove that this map has always maximal rank, allowing to get explicit recursive expression for their dimensions.

math.DG↗

arXiv2016

Quadratic and pinczon algebras

Given a symmetric non degenerated bilinear form b on a vector space V, G. Pinczon and R. Ushirobira defined a bracket {,} on the space of multilinear skewsymmetric forms on V. With this bracket, the quadratic Lie algebra structure equation on (V, b) becomes simply {â, â} = 0. We characterize similarly quadratic associative, commutative or pre-Lie structures on (V, b) by the same equation {â, â} = 0, but on different spaces of forms. These definitions extend to quadratic up to homotopy algebras and allows to describe the corresponding cohomologies.

math.QA↗

arXiv2012

Diamond cone for $\mathfrak{sl}(m,n)$

In this paper, we first study the shape algebra and the reduced shape algebra for the Lie superalgebra $\mathfrak{sl}(m,n)$. We define the quasistandard tableaux, their collection is the diamond cone for $\mathfrak{sl}(m,n)$, which is a combinatorial basis for the reduced shape algebra. We realize a bijection between the set of semistandard tableaux with shape $λ$ and the set of quasistandard tableaux with shape $μ\leqλ$, by using the 'super jeu de taquin' on skew semistandard tableaux. This gives the compatibility of the diamond cone with the natural stratification of the reduced shape algebra.

math.CO↗

arXiv2012

Pinczon Algebras

After recalling the construction of a graded Lie bracket on the space of cyclic multilinear forms on a vector space V, due to Georges Pinczon and Rosane Ushirobira, we prove this construction gives a structure of quadratic associative algebra, up to homotopy, on V. In the associative case, it is easy to refind the associated usual Hochshild cohomology. By considering restriction to a subspace or a quotient space of forms, we can present in a completely similar way the cases of quadratic commutative and quadratic Lie algebras, up to homotopy, and the corresponding Harrison and Chevalley cohomologies.

math.QA↗

arXiv2012

Diamond module for the Lie algebra $\mathfrak{so}(2n+1,\mathbb C)$

The diamond cone is a combinatorial description for a basis of an indecomposable module for the nilpotent factor $\mathfrak n$ of a semi simple Lie algebra. After N. J. Wildberger who introduced this notion, this description was achevied for $\mathfrak{sl}(n)$, the rank 2 semi-simple Lie algebras and $\mathfrak{sp}(2n)$. In the present work, we generalize these constructions to the Lie algebras $\mathfrak{so}(2n+1)$. The orthogonal semistandard Young tableaux were defined by M. Kashiwara and T. Nakashima, they form a basis for the shape algebra of $\mathfrak{so}(2n+1)$. Defining the notion of orthogonal quasistandard Young tableaux, we prove these tableaux give a basis for the diamond module for $\mathfrak{so}(2n+1)$.

math.QA↗

arXiv2012

Algèbres pré-Gerstenhaber à homotopie près

This paper is concerned by the concept of algebra up to homotopy for a structure defined by two operations $.$ and [,]. An important example of such a structure is the Gerstenhaber algebra (commutatitve and Lie). The notion of Gerstenhaber algebra up to homotopy ($G_\infty$ algebra) is known. Here, we give a definition of pre-Gerstenhaber algebra (pre-commutative and pre-Lie) allowing the construction of $\hbox{pre}G_\infty$ algebra. Given a structure of pre-commutative (Zinbiel) and pre-Lie algebra and working over the corresponding dual operads, we will give an explicit construction of the associated pre-Gerstenhaber algebra up to homotopy, this is a bicogebra (Leibniz and permutative) equipped with a codifferential which is a coderivation for the two coproducts.

math.QA↗

arXiv2010

Cohomologie de Chevalley des graphes ascendants

The space $T_{poly}(\mathbb R^d)$ of all tensor fields on $\mathbb R^d$, equipped with the Schouten bracket is a Lie algebra. The subspace of ascending tensors is a Lie subalgebra of $T_{poly}(\mathbb R^d)$. In this paper, we compute the cohomology of the adjoint representations of this algebra (in itself and $T_{poly}(\mathbb R^d)$), when we restrict ourselves to cochains defined by aerial Kontsevitch's graphs like in our previous work (Pacific J of Math, vol 229, no 2, (2007) 257-292). As in the vectorial graphs case, the cohomology is freely generated by all the products of odd wheels.

math.QA↗

arXiv2009

Le cône diamant symplectique

The diamond cone is a combinatorial description for a basis in a indecomposable module for the nilpotent factor n+ of a semi simple Lie algebra. After N.J. Wildberger who introduced this notion for sl(3), this description was achevied by N. Bel Baraka, N.J. Wildberger and D. A. for sl(n) and by B. Agrebaoui and ourselves for the rank 2 semi-simple Lie algebras. In the present work, we generalize these constructions to the Lie algebras sp(2n). The symplectic semi-standard Young tableaux were defined by C. de Concini, they form a basis for the shape algebra of sp(2n). We introduce here the notion of symplectic quasi-standard Young tableaux, these tableaux describe the diamond cone for sp(2n).

math.CO↗

arXiv2009

The spaces $\mathrm{H}^n(\mathfrak{osp}(1|2),M)$ for some weight modules $M$

We entirely compute the cohomology for a natural and large class of $\mathfrak{osp}(1|2)$ modules $M$. We study the restriction to the $\mathfrak{sl}(2)$ cohomology of $M$ and apply our results to the module $M={\mathfrak D}_{λ,μ}$ of differential operators on the super circle, acting on densities.

math.QA↗

arXiv2009

Séparation des représentations par des surgroupes quadratiques

Let $π$ be an unitary irreducible representation of a Lie group $G$. $π$ defines a moment set $I_π$, subset of the dual $\mathfrak g^*$ of the Lie algebra of $G$. Unfortunately, $I_π$ does not characterize $π$. However, we sometimes can find an overgroup $G^+$ for $G$, and associate, to $π$, a representation $π^+$ of $G^+$ in such a manner that $I_{π^+}$ characterizes $π$, at least for generic representations $π$. If this construction is based on polynomial functions with degree at most 2, we say that $G^+$ is a quadratic overgroup for $G$. In this paper, we prove the existence of such a quadratic overgroup for many different classes of $G$.

math.RT↗

arXiv2008

Diamond representations of rank two semisimple Lie algebras

The present work is a part of a larger program to construct explicit combinatorial models for the (indecomposable) regular representation of the nilpotent factor $N$ in the Iwasawa decomposition of a semi-simple Lie algebra $\mathfrak g$, using the restrictions to $N$ of the simple finite dimensional modules of $\mathfrak g$. Such a description is given in \cite{[ABW]}, for the cas $\mathfrak g=\mathfrak{sl}(n)$. Here, we give the analog for the rank 2 semi simple Lie algebras (of type $A_1\times A_1$, $A_2$, $C_2$ and $G_2$). The algebra $\mathbb C[N]$ of polynomial functions on $N$ is a quotient, called reduced shape algebra of the shape algebra for $\mathfrak g$. Basis for the shape algebra are known, for instance the so called semi standard Young tableaux (see \cite{[ADLMPPrW]}). We select among the semi standard tableaux, the so called quasi standard ones which define a kind basis for the reduced shape algebra.

math.QA↗

arXiv2008

Algèbres et cogèbres de Gerstenhaber et cohomologie de Chevalley-Harrison

The fundamental example of Gerstenhaber algebra is the space $T_{poly}({\mathbb R}^d)$ of polyvector fields on $\mathbb{R}^d$, equipped with the wedge product and the Schouten bracket. In this paper, we explicitely describe what is the enveloping $G_\infty$ algebra of a Gerstenhaber algebra $\mathcal{G}$. This structure gives us a definition of the Chevalley-Harrison cohomology operator for $\mathcal{G}$. We finally show the nontriviality of a Chevalley-Harrison cohomology group for a natural Gerstenhaber subalgebra in $T_{poly}({\mathbb R}^d)$.

math.QA↗

arXiv2005

Diamond representations of $\mathfrak{sl}(n)$

In \cite{W}, there is a graphic description of any irreducible, finite dimensional $\mathfrak{sl}(3)$ module. This construction, called diamond representation is very simple and can be easily extended to the space of irreducible finite dimensional ${\mathcal U}\_q(\mathfrak{sl}(3))$-modules. In the present work, we generalize this construction to $\mathfrak{sl}(n)$. We show this is in fact a description of the reduced shape algebra, a quotient of the shape algebra of $\mathfrak{sl}(n)$. The basis used in \cite{W} is thus naturally parametrized with the so called quasi standard Young tableaux. To compute the matrix coefficients of the representation in this basis, it is possible to use Groebner basis for the ideal of reduced Plücker relations defining the reduced shape algebra.

math.RT↗

arXiv2005

Cohomologie de Chevalley des graphes vectoriels

The space of smotth functions and vector fields on $\R^d$ is a Lie subalgebra of the (graded) Lie algebra $T\_{poly}(\R^d)$, equipped with the Scouten bracket. In this paper, we compute the cohomology of this subalgebra for the adjoint representation in $T\_{poly}(\R^d)$, restricting ourselves to the case of cochains defined with purely aerial Kontsevich's graphs, as in [AGM]. We find results which are very similar to the classical Gelfand-Fuchs and de Wilde-Lecomte one.

math.QA↗