191 Results Found

Effect algebras are the main object of study in quantum mechanics. Module measures are those measures defined on an effect algebra with values on a topological module. Let R be a topological ring and M a topological R-module. Let L be an effect algeb...

Let $S_3$ be the symmetric group on three elements. Let k be a field and $M_3\left(k\right)$ be the full matrix algebra of $3\times 3$ -matrices over k. In this paper, the $k{S}_3$ -module algebra structures on $M_3\left(k\right)$...

In this paper, we investigate the module algebra structures of $X_q\left(A_1\right)$ on the quantum polynomial algebra ${\mathbb{C}}_q[x,y,z]$. The automorphism group of ${\mathbb{C}}_q[x,y,z]$ is denoted by $\mathrm{Aut}\left({\mathbb{C}}_q[x,y,z]\right)$, and we give a complete classification of $X_q\left(A_1\right)$-module algebra struct...

The review is devoted to nonassociative algebras, rings and modules over them. The main actual and recent trends in this area are described. Works are reviewed on radicals in nonassociative rings, nonassociative algebras related with skew polynomials...

In this article, we obtain some cohomology of classical Lie algebras over an algebraically closed field of characteristic $p>h,$ where $h$ is a Coxeter number, with coefficients in simple modules. We assume that these classical Lie algebras are Lie al...

In this paper we will give some algebraic results of crossed modules of algebras.

This paper mainly considers a class of non-weight modules over the Lie algebra of the Weyl type. First, we construct the $U\left(\mathfrak{h}\right)$-free modules of rank one over the differential operator algebra. Then, we characterize the tensor products of these kind of...

In this paper, we investigate the Koszul behavior of finitely semi-graded algebras by the distributivity of some associated lattice of ideals. The Hilbert series, the Poincaré series, and the Yoneda algebra are defined for this class of algebr...

The goal of the present paper is to provide a cohomology theory and crossed modules of modified Rota–Baxter pre-Lie algebras. We introduce the notion of a modified Rota–Baxter pre-Lie algebra and its bimodule. We define a cohomology of mo...

GIM Lie algebras are the generalizations of Kac–Moody Lie algebras. However, the structures of GIM Lie algebras are more complex than the latter, so they have encountered many new difficulties to study their representation theory. In this paper...

In this paper, we construct a large class of simple restricted modules over the deformed 𝔟𝔪𝔰₃ algebra ℬ, which contain the highest weight modules and Whittaker modules. Moreover, we obtain several equivalent statements f...

In the present work, we extend to Lie modules of Banach space nest algebras a well-known characterisation of Lie ideals of (Hilbert space) nest algebras. Let $\mathcal{A}$ be a Banach space nest algebra and $\mathcal{L}$ be a weakly closed Lie $\mathcal{A}$-module. We show that there e...

In this paper, we define a cohomology theory of a modified $\lambda$-differential left-symmetric algebra. Moreover, we introduce the notion of modified $\lambda$-differential left-symmetric 2-algebras, which is the categorization of a modified $\lambda$...

In the context of cognitive radio, smart cities and Internet-of-Things, the need for advanced radio spectrum monitoring becomes crucial. However, surveillance of a wide frequency band without using extremely expensive high sampling rate devices is a...

Computer Algebra Systems (CAS) currently lack an effective auditory representation, with most existing solutions relying on screen readers that provide limited functionality. This limitation prevents blind users from fully understanding and interpret...

A research proposal on the algebraic structure, the representations and the possible applications of paraparticle algebras is structured in three modules: The first part stems from an attempt to classify the inequivalent gradings and braided group st...

In this work, we study the relationship among Lie algebraic homotopy 3-types. We investigate the categorical connection between crossed squares, internal crossed modules, 2-crossed modules, quadratic modules of Lie algebras, and simplicial and bisimp...

In this paper, we study the restricted cohomology of Lie algebras of semisimple and simply connected algebraic groups in positive characteristics with coefficients in simple restricted modules and their applications in studying the connections betwee...

In this article, we study the representation of generalized loop planar Galilean conformal algebra $W\left(\Gamma \right)$; we will determine the Harish-Chandra modules, Verma modules, Whittaker modules, and $U\left(h\right)$-free modules of $W\left(\Gamma \right)$.

The Kronecker algebra $\mathcal{K}$ is the path algebra induced by the quiver with two parallel arrows, one source and one sink (i.e., a quiver with two vertices and two arrows going in the same direction). Modules over $\mathcal{K}$ are said to be Kronecker modules. The cl...

The main purpose of this paper is to introduce and investigate the notion of Jacobi–Jordan conformal algebras. They are a generalization of Jacobi–Jordan algebras which correspond to the case in which the formal parameter $\lambda$ equals...

Hopf algebras, as a crucial generalization of groups, have a very symmetric structure and have been playing a prominent role in mathematical physics. In this paper, let H be a dual quasi-Hopf algebra which is a more general Hopf algebra structure. A....

Motivated by the relation between Albert’s Problem and irreducible modules within the class of commutative power-associative algebras, in this paper, we show some equivalences to Albert’s Problem. Furthermore, we study some properties of...

In this paper we study a monomial module M generated by an s-sequence and the main algebraic and homological invariants of the symmetric algebra of M. We show that the first syzygy module of a finitely generated module M, over any commutative Noether...

In this paper, we introduce and study two smash products $A★H$ for a left H-quasimodule algebra A over a Hopf quasigroup H over a field $\mathbb{K}$ and $B\#U$ for a coquasi U-module algebra B over a Hopf coquasigroup U, respectively. Then, we prove our dualit...

The study of algebraic structures endowed with the concept of symmetry is made possible by the link between Lie algebras and symmetric monoidal categories. This relationship between Lie algebras and symmetric monoidal categories is useful and has res...

In this paper, we mainly prove an analogue of Blattner–Montgomery’s duality-type theorem $(A⋊H)\#\widehat{H}\cong A\otimes (H\#\widehat{H})$ in the setting of an infinite-dimensional Hopf quasigroup H with a left integral, where $A⋊H$ is the smash pr...

Let $\mathcal{A}$ be a connected cochain differential graded algebra and P a finitely generated differential graded $\mathcal{A}$-module. We show that P is semi-free if it is semi-projective and it is categorically free if it is categorically projective. It can be considere...

Snake graphs are connected planar graphs consisting of a finite sequence of adjacent tiles (squares) $T_1,T_2,\dots ,T_n$. In this case, for $1\le j\le n-1$, two consecutive tiles $T_j$ and $T_{j+1}$ share exactly one edge, either the edge at the east (west...

In this paper, we generalize the homomorphisms of modules over groups and Lie algebras as being morphisms in the category of (non-symmetric) Yetter–Drinfel’d modules. These module homomorphisms play a key role in the conjecture of Yau.

The article is devoted to non-associative algebras with metagroup relations and modules over them. Their functors are studied. Satellites of functors are scrutinized. An exactness of satellite sequences and diagrams is investigated.

In this paper, we introduce two-term differential $Lei{b}_{\infty }$-conformal algebras and give characterizations of some particular classes of such two-term differential $Lei{b}_{\infty }$-conformal algebras. Furthermore, we discuss the classification of the non...

An enrichment of a category of Dieudonné modules is made by considering Yang–Baxter conditions, and these are used to obtain ring and coring operations on the corresponding Hopf algebras. Some examples of these induced structures are discussed, inclu...

In the current study, we explore digital homology modules, and investigate their fundamental properties on (pointed) digital images as one of the developments of symmetries. We also examine pointed digital Hopf spaces and base point preserving digita...

Let R be a commutative ring with identity and S a (not necessarily saturated) multiplicative subset of R. We call the ring R to be a weakly S-Noetherian ring if every S-finite proper ideal of R is an S-Noetherian R-module. In this article, we study s...

A class of nonassociative algebras is investigated with mild relations induced from metagroup structures. Modules over nonassociative algebras are studied. For a class of modules over nonassociative algebras, their extensions and splitting extensions...

In this paper, we present two different ways for computing the Kauffman bracket skein module of $S^1\times S^2$, $\mathrm{KBSM}\left(S^1\times S^2\right)$, via braids. We first extend the universal Kauffman bracket type invariant V for knots and links in the Solid Torus ST, which...

Here, we study Frobenius bimodules associated with a pair of automorphisms of an algebra and discuss their basic properties. In particular, some equivalent conditions for a finite-dimensional bimodule are proved to be Frobenius and some isomorphisms...

A generalized (3 + 1)-dimensional nonlinear wave is investigated, which defines many nonlinear phenomena in liquid containing gas bubbles. Basic theories of the natural phenomenons are usually described by nonlinear evolution equations, for example,...

Quivers (directed graphs), species (a generalization of quivers) and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to t...

Let H be a finite Hopf $C^{*}$-algebra and A a $C^{*}$-algebra of finite dimension. In this paper, we focus on the crossed product $A⋊H$ arising from the action of H on A, which is a ∗-algebra. In terms of the faithful positive Haar measure on a finite Hopf $C^{*}$-a...

The applications of digital filters based on finite field algebra codes require their conjugation with positional computing structures. Here arises the task of algorithms and structures developed for converting the positional notation codes to finite...

The aim of the paper is to construct a secondary characteristic homomorphism for Lie pseudoalgebras. The case of inner product modules is under consideration.

This study takes a detailed look at various inequalities related to the Euclidean operator radius. It examines groups of n-tuple operators, studying how they add up and multiply together. It also uncovers a unique power inequality specific to the Euc...

This paper investigates the properties of the Yang–Baxter equation, which was initially formulated in the field of theoretical physics and statistical mechanics. The equation’s framework is extended through Yang–Baxter systems, aimi...

We show that an $L_{\infty }$-algebra can be extended to a graded Hopf algebra with a codifferential. Then, we twist this extended $L_{\infty }$-algebra with a Drinfel’d twist, simultaneously twisting its modules. Taking the $L_{\infty }$-algebra as its own (...

We study the derivations of the incidence algebra $I(X,R)$, where X is a preordered set and R is an algebra over some commutative ring T. A satisfactory description of the T-module of derivations and the T-module of outer derivations of this algebra is...

We already know that the noncommutative $\mathbb{N}$-graded Noetherian algebras resemble commutative local Noetherian rings in many respects. We also know that commutative rings have the important property that every minimal acyclic complex of finitely generate...

Generalized restricted representations were introduced to facilitate the study of representations of modular Lie algebras that lack a restricted structure. At present, generalized restricted representations are defined for Lie algebras of the Cartan...

Using the first cohomology from the mirror Heisenberg–Virasoro algebra to the twisted Heisenberg algebra (as the mirror Heisenberg–Virasoro algebra module), in this paper, we determined the derivations on the mirror Heisenberg–Viras...