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Elisabeth Remm

Publications and source records attributed to Elisabeth Remm.

At least 19 recordsLinked to original sources
arXiv2026

Contact and 2-compatible Lie algebras

A $n$-dimensional Lie algebra $g=(V,\mu)$ is called $2$-compatible if it is isomorphic to a quadratic deformation of a Lie algebra $g_0=(V,\mu_0)$. By quadratic deformation we means a formal deformation $\mu_t=\mu_0+t\varphi_1+t^2\varphi_2$ where $\mu_t$ is a Lie algebra on $V \otimes K[[t]]$. It is equivalent to say that we have the following system $\sum_{i+j \leq 4} \varphi_i \circ \varphi_j= 0$. This notion naturally appears in the theory of classification of contact Lie algebras because any $(2p+1)$-dimensional contact Lie algebra is isomorphic to a quadratic deformation of the Heisenberg algebra $\mathcal{H}_{2p+1}$.

math.RA↗
arXiv2025

Anti-commutative anti-associative algebras. Acaa-algebras

Let $(A,\mu)$ be a nonassociative algebra over a field of characteristic zero. The polarization process allows us to associate two other algebras, and this correspondence is one-one, one commutative, the other anti-commutative. Assume that $\mu$ satisfies a quadratic identity $\sum_{\sigma \in \Sigma_3} a_{\sigma}\mu(\mu(x_{\sigma(i)},x_{\sigma(j)}),x_{\sigma(k)}-a_{\sigma}\mu(x_{\sigma(i)},\mu(x_{\sigma(j)},x_{\sigma(k)})=0.$ Under certain conditions, the polarization of such a multiplication determines an anticommutative multiplication also verifying a quadratic identity. Now only two identities are possible, the first is the Jacobi identity which makes this anticommutative multiplication a Lie algebra and the multiplication $\mu$ is Lie admissible, the second, less classical is given by $[[x,y],z]=[[y,z],x]=[[z,x],y].$ Such a multiplication is here called Acaa for Anticommutative and Antiassociative. We establish some properties of this type of algebras.

math.RA↗
arXiv2024

Contact forms on SL(2p)

When p is greater than 1, any left invariant Pfaffian forms on the simple Lie group SL(2p) are not contact forms. In this paper, we give a contact form on this Lie group which is invariant by the subgroup SO(2p).

math.DG↗
arXiv2024

Depolarization and distributive laws

Given a vector space with two multiplications, one commutative the other anticommutative, possibly connected by a distributive law, the depolarization principle allows to look at this triplet through a single nonassociative multiplication. This is the case of Poisson algebras. We are interested here in the cases of transposed Poisson algebras and we show in this case that depolarization cannot be done with a single multiplication. We also examine the depolarization for Hom-Lie algebras.

math.RA↗
arXiv2022

Perturbations of polynomials and applications

After reconsidering the theorem of continuity of the roots of a polynomial in terms of its coefficients in the deformation framework, we study the stability of the greater common divisor of two polynomials compared to perturbations on their roots. We apply this results to the study of deformations of a linear operator in finite dimension and in particular to the roots study of deformed matrices.

math.RA↗
arXiv2022

Anti-associative algebras

An anti-associative algebra is a nonassociative algebra whose multiplication satisfies the identity a(bc)+(ab)c=0. Such algebras are nilpotent. We describe the free anti-associative algebras with a finite number of generators. Other types of nonassociative algebras, obtained either by the process of polarization, such as Jacobi-Jordan algebras, or obtained by deformation quantization, are associated with this class of algebras. Following Markl-Remms work, we describe the operads associated with these algebra classes and in particular the cohomology complexes in relation to deformations.

math.RA↗
arXiv2022

Deformation quantization of nonassociative algebras

We investigate formal deformations of certain classes of nonassociative algebras including classes of K[{\Sigma}3]-associative algebras, Lie-admissible algebras and anti-associative algebras. In a process which is similar to Poisson algebra for the associative case we identify for each type of algebra (A, {\mu}), an algebra (A, {\mu}, {\psi}) such that the formal deformation (A[[t]], {\mu}t) is the quantization deformation of (A, {\mu}, {\psi}). The process of polarization/depolarization associate to each nonassociative algebra a couple of algebras which products are respectively commutative and skew-symmetric and is linked with the algebra obtained from the formal deformation. The anti-associative case is developed with a link with the Jacobi-Jordan algebras

math.RA↗
arXiv2020

Lie algebroids. Lie-Rinehart algebras

After recalling the notion of Lie algebroid, we construct these structures associated with contact forms or systems. We are then interested in particular classes of Lie Rinehart algebras.

math.RA↗
arXiv2020

Weakly associative and symmetric Leibniz algebras

We study a special class of weakly associative algebras: the symmetric Leibniz algebras. We describe the structure of the commutative and skew symmetric algebras associated with the polarization-depolarization principle. We also give a structure theorem for the symmetric Leibniz algebras and we study formal deformations in the context of deformation quantization.

math.RA↗
arXiv2020

Deformation quantization of non associative algebras

We study deformation quantization of nonassociative algebras whose associator satisfies some symmetric relations. This study is expanded to a larger class of nonassociative algebras includind Leibniz algebras. We apply also to this class the rule of polarization-depolarization.

math.RA↗
arXiv2019

Non-Koszulness of operads and positivity of Poincaré series

We prove that the operad of mock partially associative $n$-ary algebras is not Koszul, as conjectured by the second and the third author in 2009, and utilise the Zeilberger's algorithm for hypergeometric summation to demonstrate that non-Koszulness of that operad cannot be established by hunting for negative coefficients in the inverse of its Poincaré series.

math.KT↗
arXiv2019

Rigid Lie algebras and algebraicity

The notion of rigidity of Lie algebra is linked to the following problem: when does a Lie brackets $μ$ on a vector space g satisfy that every Lie bracket $μ_1$ sufficiently close to $μ$ is of the form $μ_1 = P.μ$ for some P in GL(g) close to the identity? A Lie algebra which satisfies the above condition will be called rigid. The most famous example is the Lie algebra sl(2,C) of square matrices of order $2$ with vanishing trace. This Lie algebra is rigid, that is any close deformation is isomorphic to it. Let us note that, for this Lie algebra, there exists a quantification of its universal algebra. This led to the definition of the famous quantum group SL(2). Another interest of studying the rigid Lie algebras is the fact that there exists, for a given dimension, only a finite number of isomorphic classes of rigid Lie algebras. So we are tempted to establish a classification. This problem has been solved up to the dimension 8. To continue in this direction, properties must be established on the structure of these algebras. One of the first results establishes an algebraicity criterion \cite{Carles}. However, the notion of algebraicity which is used is not the classical notion and it includes non-algebraic Lie algebras in the usual sense. The aim of this work is to show that a the Lie algebra is rigid, then its algebra of inner derivations is algebraic.

math.RA↗
arXiv2019

Associative and Lie deformations of Poisson algebras

Considering a Poisson algebra as a non associative algebra satisfying the Markl-Remm identity, we study deformations of Poisson algebras as deformations of this non associative algebra. This gives a natural interpretation of deformations which preserves the underlying associative structure and we study deformations which preserve the underlying Lie algebra.

math-ph↗
arXiv2019

Coadjoint orbits of Lie algebras and Cartan class

We study the coadjoint orbits of a Lie algebra in terms of Cartan class. In fact, the tangent space to a coadjoint orbit $\mathcal{O}(α)$ at the point $α$ corresponds to the characteristic space associated to the left invariant form;$α$ and its dimension is the even part of the Cartan class of $α$. We apply this remark to determine Lie algebras such that all the nontrivial orbits (nonreduced to a point) have the same dimension, in particular when this dimension is 2 or 4. We determine also the Lie algebras of dimension $2n$ or $2n+1$ having an orbit of dimension $2n$.

math.RA↗
arXiv2017

On filiform Lie algebras. Geometric and algebraic studies

A finite dimensional filiform K-Lie algebra is a nilpotent Lie algebra g whose nil index is maximal, that is equal to dim g -1. We describe necessary and sufficient conditions for a filiform algebra over an algebraically closed field of characteristic 0 to admit a contact linear form (in odd dimension) or a symplectic structure (in even dimension). If we fix a Vergne's basis, the set of filiform n-dimensional Lie algebras is a closed Zariski subset of an affine space generated by the structure constants associated with this fixed basis. Then this subset is an algebraic variety and we describe in small dimensions the algebraic components.

math.RA↗