Search Results (29)

A pair $(A,B)$ of Banach space operators is strict $(m,X)$-isometric for a Banach space operator $X\in B(\mathcal{X})$ and a positive integer m if ${▵}_{A,B}^m(X)=\left({\displaystyle \sum _{j=0}^m}\left(\begin{array}{c}m\\ j\end{array}\right)L_A^j{R}_B^j\right)(X)=0$ and ${▵}_{A,B}^{m-1}(X)\ne 0$, where $L_A$ and $R_B\in B(B(\mathcal{X}))$ are, respectively, the operators of left multiplication by A and right multiplication by B. Define operators $E_{A,B}$ and ${\mathcal{E}}_{A,B}(X)$ by $E_{A,B}=L_A{R}_B$ and ${\left({\mathcal{E}}_{A,B}(X)\right)}^n=E_{A,B}^n(X)$ for all non-negative integers n. Using little more than an algebraic argument, the following generalised version of a result relating $(m,X)$-isometric properties of pairs $(A_1,A_2)$ and $(B_1,B_2)$ to pairs $({\mathcal{E}}_{A_1,A_2}(S_1),{\mathcal{E}}_{B_1,B_2}(S_2))$ and $(E_{A_1,A_2},E_{B_1,B_2})$ is proved: if $A_i,B_i,S_i,X$ are operators in $B(\mathcal{X})$, $1\le i\le 2$ and X a quasi-affinity, then the pair $({\mathcal{E}}_{A_1,A_2}(S_1),{\mathcal{E}}_{B_1,B_2}(S_2))$ (resp., the pair $(E_{A_1,A_2},E_{B_1,B_2})$) is strict $(m,X)$-isometric for all $X\in B(\mathcal{X})$ if and only if there exist positive integers $m_i\le m$, $1\le i\le 2$ and $m=m_1+m_2-1$, and a non-zero scalar $\beta$ such that $I-{\mathcal{E}}_{\beta A_1,A_2}(S_1)$ is (strict) $m_1$-nilpotent and $I-{\mathcal{E}}_{{\displaystyle \frac{1}{\beta }}{B}_1,B_2}(S_2)$ is (strict) $m_2$-nilpotent (resp., $(\beta A_1,B_1)$ is strict $(m_1,I)$-isometric and $({\displaystyle \frac{1}{\beta }}{B}_2,A_2)$ is strict $(m_2,I)$-isometric). Full article

This paper investigates the role of solvable and nilpotent Lie algebras in the domains of cryptography and steganography, emphasizing their potential in enhancing security protocols and covert communication methods. In the context of cryptography, we explore their application in public-key infrastructure, secure data verification, and the resolution of commutator-based problems that underpin data protection strategies. In steganography, we examine how the algebraic properties of solvable Lie algebras can be leveraged to embed confidential messages within multimedia content, such as images and video, thereby reinforcing secure communication in dynamic environments. We introduce a key exchange protocol founded on the structural properties of solvable Lie algebras, offering an alternative to traditional number-theoretic approaches. The proposed Lie Exponential Diffie–Hellman Problem (LEDHP) introduces a novel cryptographic challenge based on Lie group structures, offering enhanced security through the complexity of non-commutative algebraic operations. The protocol utilizes the non-commutative nature of Lie brackets and the computational difficulty of certain algebraic problems to ensure secure key agreement between parties. A detailed security analysis is provided, including resistance to classical attacks and discussion of post-quantum considerations. The algebraic complexity inherent to solvable Lie algebras presents promising potential for developing cryptographic protocols resilient to quantum adversaries, positioning these mathematical structures as candidates for future-proof security systems. Additionally, we propose a method for secure message embedding using the Lie algebra in combination with frame deformation techniques in animated objects, offering a novel approach to steganography in motion-based media. Full article

We show that a $P{L}_{\infty }$-algebra V can be described by a nilpotent coderivation of degree $-1$ on coalgebra $P^{*}V$. Based on this result, we can generalise the result of Lada to show that every $A_{\infty }$-algebra carries a $P{L}_{\infty }$-algebra structure and every $P{L}_{\infty }$-algebra carries an $L_{\infty }$-algebra structure. In particular, we obtain a pre-Lie n-algebra structure on an arbitrary partially associative n-algebra and deduce that pre-Lie n-algebras are n-Lie admissible. Full article

Thestability of a control system is essential for its effective operation. Stability implies that small changes in input, initial conditions, or parameters do not lead to significant fluctuations in output. Various stability properties, such as inner stability, asymptotic stability, and BIBO (Bounded Input, Bounded Output) stability, are well understood for classical linear control systems in Euclidean spaces. This paper aims to thoroughly address the stability problem for a class of linear control systems defined on matrix Lie groups. This approach generalizes classical models corresponding to the latter when the group is Abelian and non-compact. It is important to note that this generalization leads to a very difficult control system, due to the complexity of the state space and the special dynamics resulting from the drift and control vectors. Several mathematical concepts help us understand and characterize stability in the classical case. We first show how to extend these algebraic, topological, and dynamical concepts from Euclidean space to a connected Lie group of matrices. Building on classical results, we identify a pathway that enables us to formulate conjectures about stability in this broader context. This problem is closely linked to the controllability and observability properties of the system. Fortunately, these properties are well established for both classes of linear systems, whether in Euclidean spaces or on Lie groups. We are confident that these conjectures can be proved in future work, initially for the class of nilpotent and solvable groups, and later for semi-simple groups. This will provide valuable insights that will facilitate, through Jouan’s Equivalence Theorem, the analysis of an important class of nonlinear control systems on manifolds beyond Lie groups. We provide an example involving a three-dimensional solvable Lie group of rigid motions in a plane to illustrate these conjectures. Full article

This paper gives a thorough characterization of chain rings with index of nilpotency 5 and residue field ${\mathbb{F}}_{p^m},$ where p represents a prime number, contributing valuable insights to the field of algebraic structures. It carefully identifies and categorizes the family of chain rings with these specifications, thereby enhancing the understanding of their properties and applications. In addition, the work offers a detailed enumeration of all chain rings containing $p^{5m}$ elements. The significance of finite chain rings is emphasized, particularly in their suitability for coding theory, which confirms their relevance in contemporary mathematical and engineering contexts. Full article

In this paper, we present some generalizations of Jacobi–Jordan algebras. More concretely, we will focus on noncommutative Jacobi–Jordan algebras, Malcev–Jordan algebras, and general Jacobi–Jordan algebras. We adapt a method, used to classify Poisson algebras, in order to classify all general Jacobi–Jordan algebras up to dimension 4, and, in particular, all noncommutative Jacobi–Jordan algebras up to dimension 4. We present the classification of Malcev–Jordan algebras up to dimension 5. As the class of Jacobi–Jordan algebras (commutative algebras that satisfy the Jacobi identity), we find that Malcev–Jordan algebras are Jordan algebras but not necessarily nilpotent. However, we show that the classification of nilpotent Malcev–Jordan algebras is sufficient to obtain the classification of the whole class. Full article

For a left module ${}_{R}M$ over a non-commutative ring R, the notion for the class of nilpotent elements ($ni{l}_R\left(M\right)$) was first introduced and studied by Sevviiri and Groenewald in 2014 (Commun. Algebra, 42, 571–577). Moreover, Armendariz and semicommutative modules are generalizations of reduced modules and $ni{l}_R\left(M\right)=0$ in the case of reduced modules. Thus, the nilpotent class plays a vital role in these modules. Motivated by this, we present the concept of nil-Armendariz modules as a generalization of reduced modules and a refinement of Armendariz modules, focusing on the class of nilpotent elements. Further, we demonstrate that the quotient module $M/N$ is nil-Armendariz if and only if N is within the nilpotent class of ${}_{R}M$. Additionally, we establish that the matrix module $M_n\left(M\right)$ is nil-Armendariz over $M_n\left(R\right)$ and explore conditions under which nilpotent classes form submodules. Finally, we prove that nil-Armendariz modules remain closed under localization. Full article

In this paper, two classes of near-Hamiltonian systems with a nilpotent center are considered: the coexistence of algebraic limit cycles and small limit cycles. For the first class of systems, there exist $2n+1$ limit cycles, which include an algebraic limit cycle and $2n$ small limit cycles. For the second class of systems, there exist $\frac{n^2+3n+2}{2}$ limit cycles, including an algebraic limit cycle and $\frac{n^2+3n}{2}$ small limit cycles. Full article

The aim of this work is to investigate the properties and classification of an interesting class of 4-dimensional 3-Hom-Lie algebras with a nilpotent twisting map $\alpha$ and eight structure constants as parameters. Derived series and central descending series are studied for all algebras in this class and are used to divide it into five non-isomorphic subclasses. The levels of solvability and nilpotency of the 3-Hom-Lie algebras in these five classes are obtained. Building upon that, all algebras of this class are classified up to Hom-algebra isomorphism. Necessary and sufficient conditions for multiplicativity of general $(n+1)$-dimensional n-Hom-Lie algebras, as well as for algebras in the considered class, are obtained in terms of the structure constants and the twisting map. Furthermore, for some algebras in this class, it is determined whether the terms of the derived and central descending series are weak subalgebras, Hom-subalgebras, weak ideals, or Hom-ideals. Full article

Cayley and nilpotent graphs arise from the interaction between graph theory and algebra and are used to visualize the structures of some algebraic objects as groups and commutative rings. On the other hand, Green and Schroll introduced Brauer graph algebras and Brauer configuration algebras to investigate the algebras of tame and wild representation types. An appropriated system of multisets (called a Brauer configuration) induces these algebras via a suitable bounded quiver (or bounded directed graph), and the combinatorial properties of such multisets describe corresponding indecomposable projective modules, the dimensions of the algebras and their centers. Undirected graphs are examples of Brauer configuration messages, and the description of the related data for their induced Brauer configuration algebras is said to be the Brauer analysis of the graph. This paper gives closed formulas for the dimensions of Brauer configuration algebras (and their centers) induced by Cayley and nilpotent graphs defined by some finite groups and finite commutative rings. These procedures allow us to give examples of Hamiltonian digraph constructions based on Cayley graphs. As an application, some quantum entangled states (e.g., Greenberger–Horne–Zeilinger and Dicke states) are described and analyzed as suitable Brauer messages. Full article

The realization of a BRST cohomology of the 4D topological gauge-affine gravity is established in terms of a superconnection formalism. The identification of fields in the quantized theory occurs directly as is usual in terms of superconnection and its supercurvature components with the double covering of the general affine group $\overline{GA}(4,\mathbb{R})$. Then, by means of an appropriate decomposition of the metalinear double-covering group $\overline{SL}(5,\mathbb{R})$ with respect to the general linear double-covering group $\overline{GL}(4,\mathbb{R})$, one can easily obtain the enlargements of the fields while remaining consistent with the BRST algebra. This leads to the descent equations, allowing us to build the observables of the theory by means of the BRST algebra constructed using a $\overline{sa}(5,\mathbb{R})$ algebra-valued superconnection. In particular, we discuss the construction of topological invariants with torsion. Full article

We deduce a non-linear commutator higher-spin (HS) symmetry algebra which encodes unitary irreducible representations of the AdS group—subject to a Young tableaux $Y(s_1,\dots ,s_k)$ with $k\ge 2$ rows—in a d-dimensional anti-de Sitter space. Auxiliary representations for a deformed non-linear HS symmetry algebra in terms of a generalized Verma module, as applied to additively convert a subsystem of second-class constraints in the HS symmetry algebra into one with first-class constraints, are found explicitly in the case of a $k=2$ Young tableaux. An oscillator realization over the Heisenberg algebra for the Verma module is constructed. The results generalize the method of constructing auxiliary representations for the symplectic $sp\left(2k\right)$ algebra used for mixed-symmetry HS fields in flat spaces [Buchbinder, I.L.; et al. Nucl. Phys. B 2012, 862, 270–326]. Polynomial deformations of the $su(1,1)$ algebra related to the Bethe ansatz are studied as a byproduct. A nilpotent BRST operator for a non-linear HS symmetry algebra of the converted constraints for $Y(s_1,s_2)$ is found, with non-vanishing terms (resolving the Jacobi identities) of the third order in powers of ghost coordinates. A gauge-invariant unconstrained reducible Lagrangian formulation for a free bosonic HS field of generalized spin $(s_1,s_2)$ is deduced. Following the results of [Buchbinder, I.L.; et al. Phys. Lett. B 2021, 820, 136470.; Buchbinder, I.L.; et al. arXiv 2022, arXiv:2212.07097], we develop a BRST approach to constructing general off-shell local cubic interaction vertices for irreducible massive higher-spin fields (being candidates for massive particles in the Dark Matter problem). A new reducible gauge-invariant Lagrangian formulation for an antisymmetric massive tensor field of spin $(1,1)$ is obtained. Full article

It is shown that for any integers $k\ge 2$, $q\ge 2k$ and $N\ge k+q+2$, there exists a real solvable Lie algebra of the first rank with a maximal torus of derivations $\mathfrak{t}$ possessing the eigenvalue spectrum $\mathrm{spec}\left(\mathfrak{t}\right)=\left(1,2,\cdots ,k,q,q+1\cdots ,N\right)$, a nilradical of the nilpotence index $N-k$ and a characteristic sequence $(N-k,1^k)$. Full article

In this paper we consider examples of complex expansion (cKdV) and perturbation (pKdV) of the Korteweg–de Vries equation (KdV) and show that these equations have a representation in the form of the zero-curvature equation. In this case, we use the Lie algebra of 4-dimensional quadratic nilpotent matrices. Moreover, it is shown that the simplest possible matrix representation of this algebra leads to the possibility of constructing a countable number of conservation laws for these equations. 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