Search Results (11)

This paper extends the notions of n-derivations and n-automorphisms from Lie algebras to nest algebras via exponential mappings. We establish necessary and sufficient conditions for triangularity, and examine the preservation of the radical, center, and ideals under these higher-order algebraic transformations. The induced group structures of n-automorphisms are explicitly characterized, including inner and non-abelian components. Several concrete examples demonstrate the applicability and depth of the theoretical findings. Full article

The symmetries of a Riemann surface $\mathsf{\Sigma }\setminus \left\{a_i\right\}$ with n punctures $a_i$ are encoded in its fundamental group ${\pi }_1\left(\mathsf{\Sigma }\right)$. Further structure may be described through representations (homomorphisms) of ${\pi }_1$ over a Lie group G as globalized by the character variety $\mathcal{C}=\mathrm{Hom}({\pi }_1,G)/G$. Guided by our previous work in the context of topological quantum computing (TQC) and genetics, we specialize on the four-punctured Riemann sphere $\mathsf{\Sigma }=S_2^{\left(4\right)}$ and the ‘space-time-spin’ group $G=S{L}_2\left(\mathbb{C}\right)$. In such a situation, $\mathcal{C}$ possesses remarkable properties: (i) a representation is described by a three-dimensional cubic surface $V_{a,b,c,d}(x,y,z)$ with three variables and four parameters; (ii) the automorphisms of the surface satisfy the dynamical (non-linear and transcendental) Painlevé VI equation (or $P_{VI}$); and (iii) there exists a finite set of 1 (Cayley–Picard)+3 (continuous platonic)+45 (icosahedral) solutions of $P_{VI}$. In this paper, we feature the parametric representation of some solutions of $P_{VI}$: (a) solutions corresponding to algebraic surfaces such as the Klein quartic and (b) icosahedral solutions. Applications to the character variety of finitely generated groups $f_p$ encountered in TQC or DNA/RNA sequences are proposed. Full article

We define a four-dimensional Lie algebra g in this paper and then prove that this Lie algebra is solvable but not nilpotent. Due to the fact that g is a Lie algebra, $\forall x,y\in g$,$[x,y]=-[y,x]$, that is, the operation $[,]$ has anti symmetry. Symmetry is a very important law, and antisymmetry is also a very important law. We studied the structure of Poisson algebras on g using the matrix method. We studied the necessary and sufficient conditions for the automorphism of this class of Lie algebras, and give the decomposition of its automorphism group by $Aut\left(g\right)=G_3{G}_1{G}_2{G}_3{G}_4{G}_7{G}_8{G}_5$, or $Aut\left(g\right)=G_3{G}_1{G}_2{G}_3{G}_4{G}_7{G}_8{G}_5{G}_6$, or $Aut\left(g\right)=G_3{G}_1{G}_2{G}_3{G}_4{G}_7{G}_8{G}_5{G}_3$, where $G_i$ is a commutative subgroup of $Aut\left(g\right)$. We give some subgroups of g’s automorphism group and systematically studied the properties of these subgroups. Full article

The main contribution of this review is to show some relevant relationships between three geometric structures on a connected Lie group G, generated by the same dynamics. Namely, Linear Control Systems, Almost Riemannian Structures, and Degenerate Dynamical Systems. These notions are generated by two ordinary differential equations on G: linear and invariant vector fields. A linear vector field on G is determined by its flow, a 1-parameter group of $Aut\left(G\right)$, the Lie group of G-automorphisms. An invariant vector field is just an element of the Lie algebra g of G. The Jouan Equivalence Theorem and the Pontryagin Maximum Principal are instrumental in this setup, allowing the extension of results from Lie groups to arbitrary manifolds for the same kind of structures which satisfy the Lie algebra finitude condition. For each structure, we present the first given examples; these examples generate the systems in the plane. Next, we introduce a general definition for these geometric structures on Euclidean spaces and G. We describe recent results of the theory. As an additional contribution, we conclude by formulating a list of open problems and challenges on these geometric structures. Since the involved dynamic comes from algebraic structures on Lie groups, symmetries are present throughout the paper. Full article

In this paper, first we establish the explicit relation between 3-derivations and 3- automorphisms of a Lie algebra using the differential and exponential map. More precisely, we show that the Lie algebra of 3-derivations is the Lie algebra of the Lie group of 3-automorphisms. Then we study the derivations and automorphisms of the standard embedding Lie algebra of a Lie triple system. We prove that derivations and automorphisms of a Lie triple system give rise to derivations and automorphisms of the corresponding standard embedding Lie algebra. Finally we compute the 3-derivations and 3-automorphisms of 3-dimensional real Lie algebras. Full article

In our investigation on quantum gravity, we introduce an infinite dimensional complex Lie algebra ${\mathfrak{g}}_{\mathsf{u}}$ that extends ${\mathbf{e}}_{\mathbf{9}}$. It is defined through a symmetric Cartan matrix of a rank 12 Borcherds algebra. We turn ${\mathfrak{g}}_{\mathsf{u}}$ into a Lie superalgebra ${\mathfrak{sg}}_{\mathsf{u}}$ with no superpartners, in order to comply with the Pauli exclusion principle. There is a natural action of the Poincaré group on ${\mathfrak{sg}}_{\mathsf{u}}$, which is an automorphism in the massive sector. We introduce a mechanism for scattering that includes decays as particular resonant scattering. Finally, we complete the model by merging the local ${\mathfrak{sg}}_{\mathsf{u}}$ into a vertex-type algebra. Full article

This work introduces a new concept, the so-called Darboux family, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie algebra automorphisms, in a relatively easy manner. The Darboux family notion can be considered as a generalisation of the Darboux polynomial for a vector field. The classification of r-matrices and solutions to classical Yang–Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail. Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix representations of certain Lie algebras with a non-trivial center is developed. Full article

This paper provides a geometric description for Lie–Hamilton systems on ${\mathbb{R}}^2$ with locally transitive Vessiot–Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of Lie–Hamilton systems on the dual of a Lie algebra to even-dimensional symplectic leaves relative to the Kirillov-Kostant-Souriau bracket. The second is a projection onto a quotient space of an automorphic Lie–Hamilton system relative to a naturally defined Poisson structure or, more generally, an automorphic Lie system with a compatible bivector field. These models give a natural framework for the analysis of Lie–Hamilton systems on ${\mathbb{R}}^2$ while retrieving known results in a natural manner. Our methods may be extended to study Lie–Hamilton systems on higher-dimensional manifolds and provide new approaches to Lie systems admitting compatible geometric structures. Full article

Multi-polar vagueness in data plays a prominent role in several areas of the sciences. In recent years, the thought of m-polar fuzzy sets has captured the attention of numerous analysts, and research in this area has escalated in the past four years. Hybrid models of fuzzy sets have already been applied to many algebraic structures, such as $BCK/BCI$ -algebras, lie algebras, groups, and symmetric groups. A symmetry of the algebraic structure, mathematically an automorphism, is a mapping of the algebraic structure onto itself that preserves the structure. This paper focuses on combining the concepts of m-polar fuzzy sets and m-polar fuzzy points to introduce a new notion called m-polar $(\alpha ,\beta )$ -fuzzy ideals in $BCK/BCI$ -algebras. The defined notion is a generalization of fuzzy ideals, bipolar fuzzy ideals, $(\alpha ,\beta )$ -fuzzy ideals, and bipolar $(\alpha ,\beta )$ -fuzzy ideals in $BCK/BCI$ -algebras. We describe the characterization of m-polar $(\in ,\in \vee q)$ -fuzzy ideals in $BCK/BCI$ -algebras by level cut subsets. Moreover, we define m-polar $(\in ,\in \vee q)$ -fuzzy commutative ideals and explore some pertinent properties. Full article

The use of automorphisms of the various Bianchi-type Lie algebras as Lie-point symmetries of the corresponding Einstein field equations entails a reduction of their order and ultimately leads to the entire solution space. When a valid reduced action principle exists, the symmetries of the configuration mini-supermetric space can also be used, in conjunction with the constraints, to provide local or non-local constants of motion. At the classical level, depending on their number, these integrals can even secure the acquisition of the entire solution space without any further solving of the dynamical equations. At the quantum level, their operator analogues can be used, along with the Wheeler–DeWitt equation, to define unique wave functions that exhibit singularity-free behavior at a semi-classical level. Full article

The François Massieu 1869 idea to derive some mechanical and thermal properties of physical systems from “Characteristic Functions”, was developed by Gibbs and Duhem in thermodynamics with the concept of potentials, and introduced by Poincaré in probability. This paper deals with generalization of this Characteristic Function concept by Jean-Louis Koszul in Mathematics and by Jean-Marie Souriau in Statistical Physics. The Koszul-Vinberg Characteristic Function (KVCF) on convex cones will be presented as cornerstone of “Information Geometry” theory, defining Koszul Entropy as Legendre transform of minus the logarithm of KVCF, and Fisher Information Metrics as hessian of these dual functions, invariant by their automorphisms. In parallel, Souriau has extended the Characteristic Function in Statistical Physics looking for other kinds of invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat and momentum as pure geometrical objects. In covariant Souriau model, Gibbs equilibriums states are indexed by a geometric parameter, the Geometric (Planck) Temperature, with values in the Lie algebra of the dynamical Galileo/Poincaré groups, interpreted as a space-time vector, giving to the metric tensor a null Lie derivative. Fisher Information metric appears as the opposite of the derivative of Mean “Moment map” by geometric temperature, equivalent to a Geometric Capacity or Specific Heat. We will synthetize the analogies between both Koszul and Souriau models, and will reduce their definitions to the exclusive Cartan “Inner Product”. Interpreting Legendre transform as Fourier transform in (Min,+) algebra, we conclude with a definition of Entropy given by a relation mixing Fourier/Laplace transforms: Entropy = (minus) Fourier_(Min,+) o Log o Laplace_(+,X). Full article