Related papers: Examples and patterns on quadratic Lie algebras

The aim of this paper is to introduce and study quadratic Hom-Lie algebras, which are Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms. We provide several constructions leading to examples and extend the double…

A deformation of the canonical algebra for kinematical observables of the quantum field theory in Minkowski space-time has been considered under the condition of Lorentz invariance. A relativistic invariant algebra obtained depends on…

Let $k$ be an arbitrary field and $d$ a positive integer. For each degenerate symmetric or antisymmetric bilinear form $M$ on $k^{d}$ we determine the structure of the Lie algebra of matrices that preserve $M$, and of the Lie algebra of…

In this paper we investigate some important basic properties of simple Hom-Lie superalgebras and show that a Hom-Lie superalgebra does not have any left or right nontrivial ideals. Moreover, we classify invariant bilinear forms on a given…

The normal form for a system of ode's is constructed from its polynomial symmetries of the linear part of the system, which is assumed to be semi-simple. The symmetries are shown to have a simple structure such as invariant function times…

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 $\mu_t=\mu_0+t\varphi_1+t^2\varphi_2$…

In this paper, we introduce a novel generalization of the classical property of algebras known as "being alternative," which we term "partially alternative." This new concept broadens the scope of alternative algebras, offering a fresh…

Leibniz algebras are a non-anticommutative version of Lie algebras. They play an important role in different areas of mathematics and physics and have attracted much attention over the last thirty years. In this paper we investigate whether…

We state criteria for a nilpotent Lie algebra $\g$ to admit an invariant metric. We use that $\g$ possesses two canonical abelian ideals $\ide(\g) \subset \mathfrak{J}(\g)$ to decompose the underlying vector space of $\g$ and then we state…

We discuss a procedure to determine finite sets $\mathcal{M}$ within the commutant of an algebraic Hamiltonian in the enveloping algebra of a Lie algebra $\mathfrak{g}$ such that their generators define a quadratic algebra. Although…

The integrability condition called shape invariance is shown to have an underlying algebraic structure and the associated Lie algebras are identified. These shape-invariance algebras transform the parameters of the potentials such as…

An algebra $L$ over a field $\Bbb F$, in which product is denoted by $[\,,\,]$, is said to be \textit{ Lie type algebra} if for all elements $a,b,c\in L$ there exist $\alpha, \beta\in \Bbb F$ such that $\alpha\neq 0$ and $[[a,b],c]=\alpha…

Let $\mf{g}$ be any finite-dimensional Lie algebra with Killling form $B$. Let $\mf{h}$ be a subalgebra of $\mf{g}$ on which the Killing form is non degenerate. Then $\mf{h}$ is reductive.

The basics of quasitriangular Hopf algebras and quantum Lie algebras are briefly reviewed, and it is shown that their properties allow the introduction of a Killing form. For quantum Lie algebras, this leads to the definitions of a Killing…

Every non-solvable and non-semisimple quadratic Lie algebra can be obtained as a double extension of a solvable quadratic Lie algebra. Thanks to a partial classification of nilpotent Lie algebras and this result, we can design different…

In this work we state a version of the double extension for homogeneous quadratic Lie super algebras that includes even and odd cases. We prove that any indecomposable, non-simple and homogeneous quadratic Lie super algebra is obtained by…

We call a linear operator on a vector space over a field Jordanable if it has a Jordan canonical form. An almost Abelian Lie algebra has only one non-vanishing Lie bracket, which is given by a linear operator. If the latter is Jordanable…

In this paper, we classify solvable quadratic Lie algebras up to dimension 6. In dimensions smaller than 6, we use the Witt decomposition given in \cite{Bou59} and a result in \cite{PU07} to obtain two non-Abelian indecomposable solvable…

As a natural generalization of ordinary Lie algebras we introduce the concept of quantum Lie algebras ${\cal L}_q(g)$. We define these in terms of certain adjoint submodules of quantized enveloping algebras $U_q(g)$ endowed with a quantum…

Non-degenerate bilinear forms over fields of characteristic 2, in particular, non-symmetric ones, are classified with respect to various equivalences, and the Lie algebras preserving them are described. Although it is known that there are…