Search Results (19)

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

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 addresses Noether’s problem for p-groups G, having an abelian normal subgroup of index p, under the condition $G^p=\{g^p:g\in G\}\le Z\left(G\right)$—the center of G. We prove that the fixed field $K\left(G\right)=K{(x\left(g\right):g\in G)}^G$ is rational over K in such cases, focusing on both the classification and structural analysis of these groups. Our results extend existing work by removing restrictive assumptions and providing a refined understanding of p-groups and their representations. 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

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

The notion of genus for finitely generated nilpotent groups was introduced by Mislin. Two finitely generated nilpotent groups Q and R belong to the same genus set $\mathcal{G}\left(Q\right)$ if and only if the two groups are nonisomorphic, but for each prime p, their p-localizations $Q_p$ and $R_p$ are isomorphic. Mislin and Hilton introduced the structure of a finite abelian group on the genus if the group Q has a finite commutator subgroup. In this study, we consider the class of finitely generated infinite nilpotent groups with a finite commutator subgroup. We construct a pullback $H_t$ from the l-equivalences $H_i\to H$ and $H_j\to H$, $t\equiv (i+j)mods$, where $s=\left|\mathcal{G}\left(H\right)\right|$, and compare its genus to that of H. Furthermore, we consider a pullback L of a direct product $G\times K$ of groups in this class. Here, we prove results on the group L and prove that its genus is nontrivial. 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

Let $\sigma =\{{\sigma }_i:i\in I\}$ be a partition of the set of all prime numbers. A subgroup H of a finite group G is said to be $\sigma$-subnormal in G if H can be joined to G by a chain of subgroups $H=H_0\subseteq H_1\subseteq \cdots \subseteq H_n=G$ where, for every $j=1,\cdots ,n$, $H_{j-1}$ is normal in $H_j$ or $H_j/Cor{e}_{H_j}\left(H_{j-1}\right)$ is a ${\sigma }_i$-group for some $i\in I$. Let B be a subgroup of a soluble group G normalising the ${\mathfrak{N}}_{\sigma }$-residual of every non-$\sigma$-subnormal subgroup of G, where ${\mathfrak{N}}_{\sigma }$ is the saturated formation of all $\sigma$-nilpotent groups. We show that B normalises the ${\mathfrak{N}}_{\sigma }$-residual of every subgroup of G if G does not have a section that is $\sigma$-residually critical. Full article

The idea of a canonical ensemble from Gibbs has been extended by Jean-Marie Souriau for a symplectic manifold where a Lie group has a Hamiltonian action. A novel symplectic thermodynamics and information geometry known as “Lie group thermodynamics” then explains foliation structures of thermodynamics. We then infer a geometric structure for heat equation from this archetypal model, and we have discovered a pure geometric structure of entropy, which characterizes entropy in coadjoint representation as an invariant Casimir function. The coadjoint orbits form the level sets on the entropy. By using the KKS 2-form in the affine case via Souriau’s cocycle, the method also enables the Fisher metric from information geometry for Lie groups. The fact that transverse dynamics to these symplectic leaves is dissipative, whilst dynamics along these symplectic leaves characterize non-dissipative phenomenon, can be used to interpret this Lie group thermodynamics within the context of an open system out of thermodynamics equilibrium. In the following section, we will discuss the dissipative symplectic model of heat and information through the Poisson transverse structure to the symplectic leaf of coadjoint orbits, which is based on the metriplectic bracket, which guarantees conservation of energy and non-decrease of entropy. Baptiste Coquinot recently developed a new foundation theory for dissipative brackets by taking a broad perspective from non-equilibrium thermodynamics. He did this by first considering more natural variables for building the bracket used in metriplectic flow and then by presenting a methodical approach to the development of the theory. By deriving a generic dissipative bracket from fundamental thermodynamic first principles, Baptiste Coquinot demonstrates that brackets for the dissipative part are entirely natural, just as Poisson brackets for the non-dissipative part are canonical for Hamiltonian dynamics. We shall investigate how the theory of dissipative brackets introduced by Paul Dirac for limited Hamiltonian systems relates to transverse structure. We shall investigate an alternative method to the metriplectic method based on Michel Saint Germain’s PhD research on the transverse Poisson structure. We will examine an alternative method to the metriplectic method based on the transverse Poisson structure, which Michel Saint-Germain studied for his PhD and was motivated by the key works of Fokko du Cloux. In continuation of Saint-Germain’s works, Hervé Sabourin highlights the, for transverse Poisson structures, polynomial nature to nilpotent adjoint orbits and demonstrated that the Casimir functions of the transverse Poisson structure that result from restriction to the Lie–Poisson structure transverse slice are Casimir functions independent of the transverse Poisson structure. He also demonstrated that, on the transverse slice, two polynomial Poisson structures to the symplectic leaf appear that have Casimir functions. The dissipative equation introduced by Lindblad, from the Hamiltonian Liouville equation operating on the quantum density matrix, will be applied to illustrate these previous models. For the Lindblad operator, the dissipative component has been described as the relative entropy gradient and the maximum entropy principle by Öttinger. It has been observed then that the Lindblad equation is a linear approximation of the metriplectic equation. Full article

Let $\sigma =\{{\sigma }_i:i\in I\}$ be a partition of the set $\mathbb{P}$ of all prime numbers and let G be a finite group. We say that G is $\sigma$-primary if all the prime factors of $\left|G\right|$ belong to the same member of $\sigma$. G is said to be $\sigma$-soluble if every chief factor of G is $\sigma$-primary, and G is $\sigma$-nilpotent if it is a direct product of $\sigma$-primary groups. It is known that G has a largest normal $\sigma$-nilpotent subgroup which is denoted by $F_{\sigma }\left(G\right)$. Let n be a non-negative integer. The n-term of the $\sigma$-Fitting series of G is defined inductively by $F_0\left(G\right)=1$, and $F_{n+1}\left(G\right)/F_n\left(G\right)=F_{\sigma }(G/F_n\left(G\right))$. If G is $\sigma$-soluble, there exists a smallest n such that $F_n\left(G\right)=G$. This number n is called the $\sigma$-nilpotent length of G and it is denoted by $l_{\sigma }\left(G\right)$. If $\mathfrak{F}$ is a subgroup-closed saturated formation, we define the $\sigma$-$\mathfrak{F}$-length$n_{\sigma }(G,\mathfrak{F})$ of G as the $\sigma$-nilpotent length of the $\mathfrak{F}$-residual $G^{\mathfrak{F}}$ of G. The main result of the paper shows that if A is a maximal subgroup of G and G is a $\sigma$-soluble, then $n_{\sigma }(A,\mathfrak{F})=n_{\sigma }(G,\mathfrak{F})-i$ for some $i\in \{0,1,2\}$. Full article

This paper surveys results related to well-known works of B. Plotkin and V. Remeslennikov on the edge of algebra, logic and geometry. We start from a brief review of the paper and motivations. The first sections deal with model theory. In the first part of the second section we describe the geometric equivalence, the elementary equivalence, and the isotypicity of algebras. We look at these notions from the positions of universal algebraic geometry and make emphasis on the cases of the first order rigidity. In this setting Plotkin’s problem on the structure of automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of categories is pretty natural and important. The second part of the second section is dedicated to particular cases of Plotkin’s problem. The last part of the second section is devoted to Plotkin’s problem for automorphisms of the group of polynomial symplectomorphisms. This setting has applications to mathematical physics through the use of model theory (non-standard analysis) in the studying of homomorphisms between groups of symplectomorphisms and automorphisms of the Weyl algebra. The last sections deal with algorithmic problems for noncommutative and commutative algebraic geometry.The first part of it is devoted to the Gröbner basis in non-commutative situation. Despite the existence of an algorithm for checking equalities, the zero divisors and nilpotency problems are algorithmically unsolvable. The second part of the last section is connected with the problem of embedding of algebraic varieties; a sketch of the proof of its algorithmic undecidability over a field of characteristic zero is given. Full article

In this paper, we mainly count the number of subgroup chains of a finite nilpotent group. We derive a recursive formula that reduces the counting problem to that of finite p-groups. As applications of our main result, the classification problem of distinct fuzzy subgroups of finite abelian groups is reduced to that of finite abelian p-groups. In particular, an explicit recursive formula for the number of distinct fuzzy subgroups of a finite abelian group whose Sylow subgroups are cyclic groups or elementary abelian groups is given. Full article

For a non-empty class of groups $\mathcal{L}$, a finite group $G=AB$ is said to be an $\mathcal{L}$-connected product of the subgroups A and B if $⟨a,b⟩\in \mathcal{L}$ for all $a\in A$ and $b\in B$. In a previous paper, we prove that, for such a product, when $\mathcal{L}=\mathcal{S}$ is the class of finite soluble groups, then $[A,B]$ is soluble. This generalizes the theorem of Thompson that states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper, our result is applied to extend to finite groups previous research about finite groups in the soluble universe. In particular, we characterize connected products for relevant classes of groups, among others, the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Additionally, we give local descriptions of relevant subgroups of finite groups. Full article

Given a positive integer n, a finite group G is called quasi-core-n if $⟨x⟩/{⟨x⟩}_G$ has order at most n for any element x in G, where ${⟨x⟩}_G$ is the normal core of $⟨x⟩$ in G. In this paper, we investigate the structure of finite quasi-core-p p-groups. We prove that if the nilpotency class of a quasi-core-p p-group is $p+m$ , then the exponent of its commutator subgroup cannot exceed $p^{m+1}$ , where p is an odd prime and m is non-negative. If $p=3$ , we prove that every quasi-core-3 3-group has nilpotency class at most 5 and its commutator subgroup is of exponent at most 9. We also show that the Frattini subgroup of a quasi-core-2 2-group is abelian. Full article

A subgroup H of a finite group G is said to be weakly $\mathcal{H}$ -embedded in G if there exists a normal subgroup T of G such that $H^G=HT$ and $H\cap T\in \mathcal{H}\left(G\right)$ , where $H^G$ is the normal closure of H in G, and $\mathcal{H}\left(G\right)$ is the set of all $\mathcal{H}$ -subgroups of G. In the recent research, Asaad, Ramadan and Heliel gave new characterization of p-nilpotent: Let p be the smallest prime dividing $\left|G\right|$ , and P a non-cyclic Sylow p-subgroup of G. Then G is p-nilpotent if and only if there exists a p-power d with $1<d<\left|P\right|$ such that all subgroups of P of order d and $pd$ are weakly $\mathcal{H}$ -embedded in G. As new applications of weakly $\mathcal{H}$ -embedded subgroups, in this paper, (1) we generalize this result for general prime p and get a new criterion for p-supersolubility; (2) adding the condition “ $N_G\left(P\right)$ is p-nilpotent”, here $N_G\left(P\right)=\{g\in G|P^g=P\}$ is the normalizer of P in G, we obtain p-nilpotence for general prime p. Moreover, our tool is the weakly $\mathcal{H}$ -embedded subgroup. However, instead of the normality of $H^G=HT$ , we just need $HT$ is S-quasinormal in G, which means that $HT$ permutes with every Sylow subgroup of G. Full article