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On this work we study associative triple systems of the second kind. We show that for simple triple systems the automorphism group scheme is isomorphic to the automorphism group scheme of the 3-graded associative algebra with involution constructed by Loos. This result will allow us to prove our main result which is a complete classification up to isomorphism of the gradings of structurable algebras. [ABSTRACT FROM AUTHOR]

The notions of the Novikov deformation of a commutative associative algebra and the corresponding classical limit are introduced. We show such a classical limit belongs to a subclass of transposed Poisson algebras, and hence the Novikov deformation is defined to be the quantization of the corresponding transposed Poisson algebra. As a direct consequence, we revisit the relationship between transposed Poisson algebras and Novikov–Poisson algebras due to the fact that there is a natural Novikov deformation of the commutative associative algebra in a Novikov–Poisson algebra. Hence all transposed Poisson algebras of Novikov–Poisson type, including unital transposed Poisson algebras, can be quantized. Finally, we classify the quantizations of 2-dimensional complex transposed Poisson algebras in which the Lie brackets are non-abelian up to equivalence. [ABSTRACT FROM AUTHOR]

The paper concerns perfect diassociative algebras and their implications to the theory of central extensions. It is first established that perfect diassociative algebras have strong ties with universal central extensions. Then, using a known characterization of the multiplier in terms of a free presentation, we obtain a special cover for perfect diassociative algebras, as well as some of its properties. The subsequent results connect and build on the previous topics. For the final theorem, we invoke an extended Hochschild–Serre-type spectral sequence to show that, for a perfect diassociative algebra, its cover is perfect and has trivial multiplier. As an important consequence, we obtain the entire theory for associative algebras as a special case of diassociative algebras. We conclude with a concrete example. [ABSTRACT FROM AUTHOR]

We introduce and study linear soficity for Lie algebras, modeled after linear soficity in associative algebras. We introduce equivalent definitions of linear soficity, one involving metric ultraproducts and the other involving almost representations. We prove that any Lie algebra of subexponential growth is linearly sofic. We also prove that a Lie algebra over a field of characteristic 0 is linearly sofic if and only if its universal enveloping algebra is linearly sofic. As examples, we give explicit families of almost representations for the Witt and Virasoro algebras. Finally, we also show that the restricted universal enveloping algebra of a restricted linearly sofic Lie algebra is also linearly sofic. [ABSTRACT FROM AUTHOR]

AbstractThis article is devoted to the classification of anti-dendriform algebras that are associated with associativity. In particular, the paper is devoted to classifying anti-dendriform algebras associated with null-filiform associative algebras and three-dimensional algebras. It is known that any finite-dimensional associative algebra without a nonzero idempotent element is nilpotent. If an associative algebra has a nonzero idempotent element, then there does not exist a compatible anti-dendriform algebra structure associated with associative algebras. [ABSTRACT FROM AUTHOR]

We present a comprehensive study of nearly associative algebras. Our research shows that these algebras are power associative Lie-admissible for which the related Lie algebra is solvable, and are also Jordan-admissible. Furthermore, we describe the features of finite-dimensional nearly associative algebras that are nilpotent. In addition, we define the concepts of radical and semisimplicity for this type of algebras. We give a characterization of semisimple nearly associative algebras and prove the Wedderburn Principal Theorem for them. Lastly, we deal with quadratic nearly associative algebras. We provide a characterization and then an inductive description of them through the process of double extension, enriching our understanding of the structural intricacies of this class of algebras. [ABSTRACT FROM AUTHOR]

Given a nonsymmetric operad 풪, we first construct two new nonsymmetric operads 풪comp and 풪Dend. These operads are respectively useful to study compatible and split Loday-algebras. As an application of the operad 풪comp, we show that the cohomology of a compatible associative algebra carries a Gerstenhaber structure. We give an application of the operad 풪Dend to dendriform algebras and find generalizations to other Loday-algebras. In the end, we construct another operad Fam(풪Ω)Dend to study dendriform-family algebras recently introduced in the literature. We also define and study homotopy dendriform-family algebras. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of crossed modules of associative conformal algebras, two-term strongly homotopy associative conformal algebras, and discuss the relationship between them and the third Hochschild cohomology of associative conformal algebras. We classify the non-abelian extensions by introducing the non-abelian cohomology. We show that non-abelian extensions of an associative conformal algebra can be viewed as Maurer–Cartan elements of a suitable differential graded Lie algebra, and prove that there is a one-one correspondence between the Deligne groupoid of this differential graded Lie algebra and the non-abelian cohomology. For any two associative conformal algebras A and B, we provide the condition that a pair of automorphisms in Aut(A) × Aut(B) can be extended to an automorphisms of the non-abelian extension algebra of B by A, and give the fundamental sequence of Wells in the context of associative conformal algebras. [ABSTRACT FROM AUTHOR]

This paper investigates Rota–Baxter algebras of of arbitrary weight, that is, associative algebras endowed with Rota-Baxter operators of arbitrary weight, from an operadic viewpoint. Denote by λ RBA the operad of Rota-Baxter associative algebras of weight λ . A homotopy cooperad is explicitly constructed, which can be seen as the Koszul dual of λ RBA as it is proven that the cobar construction of this homotopy cooperad is exactly the minimal model of λ RBA . This enables us to introduce the notion of homotopy Rota-Baxter algebras. The deformation complex of a Rota-Baxter algebra and the underlying L ∞ -algebra structure over it are exhibited as well. [ABSTRACT FROM AUTHOR]

To any associative algebra A is associated a numerical sequence c_n^{\delta }(A), n\ge 1, called the sequence of proper central codimensions of A. It gives information on the growth of the proper central polynomials of the algebra. If A is a PI-algebra over a field of characteristic zero it has been recently shown that such a sequence either grows exponentially or is polynomially bounded. Here we classify, up to PI-equivalence, the algebras A for which the sequence c_n^{\delta }(A), n\ge 1, has almost polynomial growth. Then we face a similar problem in the setting of group-graded algebras and we obtain a classification also in this case when the corresponding sequence of proper central codimensions has almost polynomial growth. [ABSTRACT FROM AUTHOR]

Quaternion algebra ℍ is a noncommutative associative algebra. In recent years, quaternionic Fourier analysis has received increasing attention due to its applications in signal analysis and image processing. This paper addresses conjugate phase retrieval problem in the quaternion Euclidean space ℍ M with M ≥ 2 . Write ℂ η = { ξ : ξ = ξ 0 + β ⁢ η , ξ 0 , β ∈ ℝ } for η ∈ { i , j , k } . We remark that not only ℂ η M -vectors cannot allow traditional conjugate phase retrieval in ℍ M , but also ℂ i M ∪ ℂ j M -complex vectors cannot allow phase retrieval in ℍ M . We are devoted to conjugate phase retrieval of ℂ i M ∪ ℂ j M -complex vectors in ℍ M , where "conjugate" is not the traditional conjugate. We introduce the notions of conjugation, maximal commutative subset and conjugate phase retrieval. Using the phase lifting techniques, we present some sufficient conditions on complex vectors allowing conjugate phase retrieval. And some examples are also provided to illustrate the generality of our theory. [ABSTRACT FROM AUTHOR]

Let U be an associative unital algebra containing a non-trivial idempotent e. We consider U as a superalgebra whose Z 2 -grading is induced by e. This paper aims to describe Lie superderivations of U . In particular, we characterize the general form of Lie superderivations of U and apply it to present the necessary and sufficient conditions for a Lie superderivation on U to be proper. Similar results have been presented for triangular algebras as superalgebras, wherein their Z 2 -grading is also obtained concerning standard idempotent. The main result is subsequently applied to full matrix algebras and upper triangular matrix algebras. [ABSTRACT FROM AUTHOR]

Mathematics is the foundational discipline for all sciences, engineering, and technology, and the pursuit of normed division algebras in all finite dimensions represents a paramount mathematical objective. In the quest for a real three-dimensional, normed, associative division algebra, Hamilton discovered quaternions, constituting a non-commutative division algebra of quadruples. Subsequent investigations revealed the existence of only four division algebras over reals, each with dimensions 1, 2, 4, and 8. This study transcends such limitations by introducing generalized hypercomplex numbers extending across all dimensions, serving as extensions of traditional complex numbers. The space formed by these numbers constitutes a non-distributive normed division algebra extendable to all finite dimensions. The derivation of these extensions involves the definitions of two new π-periodic functions and a unified multiplication operation, designated as spherical multiplication, that is fully compatible with the existing multiplication structures. Importantly, these new hypercomplex numbers and their associated algebras are compatible with the existing real and complex number systems, ensuring continuity across dimensionalities. Most importantly, like the addition operation, the proposed multiplication in all dimensions forms an Abelian group while simultaneously preserving the norm. In summary, this study presents a comprehensive generalization of complex numbers and the Euler identity in higher dimensions, shedding light on the geometric properties of vectors within these extended spaces. Finally, we elucidate the practical applications of the proposed methodology as a viable alternative for expressing a quantum state through the multiplication of specified quantum states, thereby offering a potential complement to the established superposition paradigm. Additionally, we explore its utility in point cloud image processing. [ABSTRACT FROM AUTHOR]

We study fibration category structures induced by enrichments over symmetric monoidal categories that are also fibration categories. Let V be a monoidal category that is also a fibration category. Assuming that V has an interval object, we demonstrate that the fibration category structure on V can be transferred to any V -enriched category through corepresentable functors, provided certain power objects exist. Furthermore, we extend this result to its G-equivariant version for a group G, showing that, under mild conditions, the category of G-objects in a V -enriched category admits a (nontrivial) fibration category structure. We also show that several categories of topological algebras and associative algebras, along with their G-equivariant analogues, can be structured as fibration categories obtained through this method. Finally, we present some applications of these results, including the recovery of existing findings related to (equivariant) K-theory and E-theory of operator algebras. [ABSTRACT FROM AUTHOR]

The approach for Poisson bialgebras characterized by Manin triples with respect to the invariant bilinear forms on both the commutative associative algebras and the Lie algebras is not available for giving a bialgebra theory for transposed Poisson algebras. Alternatively, we consider Manin triples with respect to the commutative 2-cocycles on the Lie algebras instead. Explicitly, we first introduce the notion of anti-pre-Lie bialgebras as the equivalent structure of Manin triples of Lie algebras with respect to the commutative 2-cocycles. Then we introduce the notion of anti-pre-Lie Poisson bialgebras, characterized by Manin triples of transposed Poisson algebras with respect to the bilinear forms which are invariant on the commutative associative algebras and commutative 2-cocycles on the Lie algebras, giving a bialgebra theory for transposed Poisson algebras. Finally the coboundary cases and the related structures such as analogues of the classical Yang–Baxter equation and -operators are studied. [ABSTRACT FROM AUTHOR]

Let D be a Noetherian infinite integral domain, denote by M 2 (D) and by s l 2 (D) the 2 × 2 matrix algebra and the Lie algebra of the traceless matrices in M 2 (D) , respectively. In this paper we study the weak polynomial identities for the natural grading by the cyclic group Z 2 of order 2 on M 2 (D) and on s l 2 (D). We describe a finite basis of the graded polynomial identities for the pair (M 2 (D) , s l 2 (D)). Moreover we prove that the ideal of the graded identities for this pair satisfies the Specht property, that is every ideal of graded identities of pairs (associative algebra, Lie algebra), satisfying the graded identities for (M 2 (D) , s l 2 (D)) , is finitely generated. The polynomial identities for M 2 (D) are known if D is any field of characteristic different from 2. The identities for the Lie algebra s l 2 (D) are known when D is an infinite field. The identities for the pair we consider were first described by Razmyslov when D is a field of characteristic 0, and afterwards by the second author when D is an infinite field. The graded identities for the pair (M 2 (D) , g l 2 (D)) were also described, by Krasilnikov and the second author. In order to obtain these results we use certain graded analogues of the generic matrices, and also techniques developed by G. Higman concerning partially well ordered sets. [ABSTRACT FROM AUTHOR]

For any vertex operator algebra V , finite automorphism g of V of order T and m , n ∈ (1 / T) Z + , we construct a family of associative algebras A g , n (V) and A g , n (V) − A g , m (V) -bimodules A g , n , m (V) from the point of view of representation theory. We prove that the algebra A g , n (V) is identical to the algebra A g , n (V) constructed by Dong, Li and Mason, and that the bimodule A g , n , m (V) is identical to A g , n , m (V) which was constructed by Dong and Jiang. We also prove that the A g , n (V) − A g , m (V) -bimodule A g , n , m (V) is isomorphic to U (V [ g ]) n − m / U (V [ g ]) n − m − m − 1 / T , where U (V [ g ]) k is the subspace of degree k of the (1 / T) Z -graded universal enveloping algebra U (V [ g ]) of V with respect to g and U (V [ g ]) k l is some subspace of U (V [ g ]) k. And we show that all these bimodules A g , n , m (V) can be defined in a simpler way. [ABSTRACT FROM AUTHOR]

This article explores the automorphisms of a Chevalley group of type G2 over a commutative ring R. The authors demonstrate that every automorphism of this group, when R is generated by the invertible elements and the ideal 2R, can be expressed as a combination of ring and inner automorphisms. The paper offers a historical overview of the study of automorphisms of classical groups and Chevalley groups, as well as the methods employed in previous research. The authors introduce definitions and main theorems related to Chevalley groups and their automorphisms. The text focuses on automorphisms of the group Gad(G2, R) and their properties, defining ring automorphisms and inner automorphisms, and establishing standard automorphisms as compositions of these two types. The primary objective is to prove that any automorphism of the group Gad(G2, R) is standard. The text also covers definitions and theorems concerning the localization of rings and modules, isomorphisms of Chevalley groups over fields, and the characteristic subgroup Ead(G2, R) in Gad(G2, R). The main theorem's proof is outlined in several steps, including the embedding of the ring R into a product of its localizations and the mapping of elements under conjugation by an element of Gad(G2, S). Additionally, a lemma is proven that demonstrates the mapping of matrices under conjugation by an element of Gad(G2, S) [Extracted from the article]

The article titled "Gradings of Galois Extensions" explores the concept of gradings in finite field extensions, specifically focusing on fine gradings. The authors investigate fine gradings of Kummer extensions and describe all possible gradings of Kummer extensions. They also examine a wider class of Galois extensions that admit fine gradings and discuss their applications in computing Galois groups. The article provides definitions, propositions, and lemmas to support their findings. The text discusses various cases and propositions related to Kummer extensions and Galois extensions, providing proofs and explanations for each proposition. It also discusses nonstandard gradings of Kummer extensions and provides criteria for their existence. The text concludes by discussing the coarsening of a fine grading of a Galois extension. Overall, the text provides a detailed analysis of fine gradings in field extensions and their implications. [Extracted from the article]

In this paper, necessary and sufficient conditions on a group algebra of a finitely generated group G over a finite field K are determined such that the set of derivations of the group algebra form an associative K -algebra. The derivations of K G form a nontrivial associative K -algebra if and only if K has characteristic 2 and G is the direct product of a finite abelian group of odd order with either a cyclic 2-group or an infinite cyclic group. In this special case, the Jacobson radical of the resulting K -algebra is determined. [ABSTRACT FROM AUTHOR]

A relatively new and rapidly expanding area of mathematics research is the study of graphs' spectral properties. Spectral graph theory plays a very important role in understanding certifiable applications such as cryptography, combinatorial design, and coding theory. Nonassociative algebras, loop groups, and quasigroups are the generalizations of associative algebra. Many studies have focused on the spectral properties of simple graphs connected to associative algebras like finite groups and rings, but the same research direction remains unexplored for loop groups and quasigroups. Eigenvalue analysis, subgraph counting, matrix representation, and the combinatorial approach are key techniques and methods in our work. The main purpose of this paper is to characterize finite quasigroups with the help of relatively prime graphs. Moreover, we investigate the structural and spectral properties of these graphs associated with finite quasigroups in the forms of star graphs, eigenvalues, connectivity, girth, clique, and chromatic number. [ABSTRACT FROM AUTHOR]

This paper explores geometric circulant matrices whose entries are Chebyshev polynomials of the first or second kind. Motivated by our previous research on r − circulant matrices and Chebyshev polynomials, we focus on calculating the Frobenius norm and deriving estimates for the spectral norm bounds of these matrices. Our analysis reveals that this approach yields notably improved results compared to previous methods. To validate the practical significance of our research, we apply it to existing studies on geometric circulant matrices involving the generalized k − Horadam numbers. The obtained results confirm the effectiveness and utility of our proposed approach. [ABSTRACT FROM AUTHOR]

Motivated by the concept of matching Rota-Baxter algebras arising from polarized associative Yang-Baxter equations and Volterra integral equations, we introduce the notion of a matching Rota-Baxter system, which generalizes the Rota-Baxter system proposed by Brzeziński. We show that this notion is also related to Yang-Baxter pairs and to matching pre-Lie algebras. We then modify the definition of matching Rota-Baxter systems by adding a curvature term, and make a connection with matching pre-Lie algebras and with compatible associative algebras. Furthermore, we study matching Rota-Baxter systems on a dendriform algebra and show how they induce matching quadri-algebra structures. Finally, we give a linear basis of free matching Rota-Baxter system by Gröbner-Shirshov bases methods. [ABSTRACT FROM AUTHOR]

We introduce the variety B sup of bicommutative superalgebras over an arbitrary field of characteristic different from 2. The variety consists of all nonassociative Z 2 -graded algebras satisfying the polynomial super-identities of super- left- and right-commutativity x (y z) = (− 1) x ‾ y ‾ y (x z) and (x y) z = (− 1) y ‾ z ‾ (x z) y , where u ‾ ∈ { 0 , 1 } is the parity of the homogeneous element u. We present an explicit construction of the free bicommutative superalgebras, find their bases as vector spaces and show that they share many properties typical for ordinary bicommutative algebras and super-commutative associative superalgebras. In particular, in the case of free algebras of finite rank we compute the Hilbert series and find explicitly its coefficients. As a consequence we give a formula for the codimension sequence. We establish an analogue of the classical Hilbert Basissatz for two-sided ideals. We see that the Gröbner-Shirshov bases of these ideals are finite, the Gelfand-Kirillov dimensions of finitely generated bicommutative superalgebras are nonnegative integers and the Hilbert series of finitely generated graded bicommutative superalgebras are rational functions. Concerning problems studied in the theory of varieties of algebraic systems, we prove that the variety of bicommutative superalgebras satisfies the Specht property. In the case of characteristic 0 we compute the sequence of cocharacters. [ABSTRACT FROM AUTHOR]

Hom-Lie algebras having non-invertible twist maps in their centroids are studied. Central extensions of Hom-Lie algebras having these properties are obtained and shown how the same properties are preserved. Conditions are given so that the produced central extension has an invariant metric with respect to its Hom-Lie product making its twist map self-adjoint when the original Hom-Lie algebra has such a metric. This work is focused on algebras with these properties and following Benayadi and Makhlouf we call them quadratic Hom-Lie algebras. It is shown how a quadratic Hom-Lie algebra gives rise to a quadratic Lie algebra and that the Lie algebra associated to the given Hom-Lie central extension is a Lie algebra central extension of it. It is also shown that if the Hom-Lie product is not a Lie product, there exists a non-abelian algebra, which is in general non-associative too, the commutator of whose product is precisely the Hom-Lie product of the Hom-Lie central extension. Moreover, the algebra whose commutator realizes this Hom-Lie product is shown to be simple if the associated Lie algebra is nilpotent. Non-trivial examples are provided. [ABSTRACT FROM AUTHOR]

We introduce an associative algebra A ∞ (V) using infinite matrices with entries in a grading-restricted vertex algebra V such that the associated graded space Gr (W) = ∐ n ∈ ℕ Gr n (W) of a filtration of a lower-bounded generalized V -module W is an A ∞ (V) -module satisfying additional properties (called a nondegenerate graded A ∞ (V) -module). We prove that a lower-bounded generalized V -module W is irreducible or completely reducible if and only if the nondegenerate graded A ∞ (V) -module Gr (W) is irreducible or completely reducible, respectively. We also prove that the set of equivalence classes of the lower-bounded generalized V -modules is in bijection with the set of the equivalence classes of nondegenerate graded A ∞ (V) -modules. For N ∈ ℕ , there is a subalgebra A N (V) of A ∞ (V) such that the subspace Gr N (W) = ∐ n = 0 N Gr n (W) of Gr (W) is an A N (V) -module satisfying additional properties (called a nondegenerate graded A N (V) -module). We prove that A N (V) are finite-dimensional when V is of positive energy (CFT type) and C 2 -cofinite. We prove that the set of the equivalence classes of lower-bounded generalized V -modules is in bijection with the set of the equivalence classes of nondegenerate graded A N (V) -modules. In the case that V is a Möbius vertex algebra and the differences between the real parts of the lowest weights of the irreducible lower-bounded generalized V -modules are less than or equal to N ∈ ℕ , we prove that a lower-bounded generalized V -module W of finite length is irreducible or completely reducible if and only if the nondegenerate graded A N (V) -module Gr N (W) is irreducible or completely reducible, respectively. [ABSTRACT FROM AUTHOR]

In this article, we mainly study the products of commutator ideals of Lie-admissible algebras such as Novikov algebras, bicommutative algebras, and assosymmetric algebras. More precisely, we first study the properties of the lower central chains for Novikov algebras and bicommutative algebras. Then we show that for every Lie nilpotent Novikov algebra or Lie nilpotent bicommutative algebra A ,the ideal of A generated by the set { a b − b a ∣ a , b ∈ A } is nilpotent. Finally, we study properties of the lower central chains for assosymmetric algebras, study the products of commutator ideals of assosymmetric algebras and show that the products of commutator ideals have a similar property as that for associative algebras. [ABSTRACT FROM AUTHOR]

We extend the theory of Matsuo algebras, which are certain non-associative algebras related to 3-transposition groups, to characteristic 2. The decompositions of our algebras are now induced by nilpotent elements associated to lines in the corresponding Fischer space, rather than idempotent elements associated to points. For many 3-transposition groups, this still gives rise to a Z / 2 Z -graded fusion law, and we provide a complete classification of when this occurs. In one particular small case, arising from the 3-transposition group Sym (4) , the fusion law is even stronger, and the resulting Miyamoto group is an algebraic group G a 2 ⋊ G m. [ABSTRACT FROM AUTHOR]

An associative central simple algebra is a form of a matrix algebra, because a maximal étale subalgebra acts on the algebra faithfully by left and right multiplication. In an attempt to extract and isolate the full potential of this point of view, we study nonassociative algebras whose nucleus contains an étale subalgebra bi-acting faithfully on the algebra. These algebras, termed semiassociative, are shown to be the forms of skew matrix algebras, which we are led to define and investigate. Semiassociative algebras modulo skew matrix algebras compose a Brauer monoid, which contains the Brauer group of the field as a unique maximal subgroup. [ABSTRACT FROM AUTHOR]

In this paper we use the classical Wedderburn's Kronecker Factorization Theorem to prove that category of bimodules over B and the category of bimodules over Mn(B) are equivalent, where B is some unital associative algebra. In addition to this, we classify the irreducible bimodules over Mn(F). [ABSTRACT FROM AUTHOR]

In this work, we consider genetic algebras, generated by Volterra quadratic stochastic operators, and establish relation between regularity of Volterra operators and associativity of corresponding genetic algebra. [ABSTRACT FROM AUTHOR]

One of the questions investigated in deformation theory is to determine to which algebras can a given associative algebra be deformed. In this paper we investigate a different but related question, namely: for a given associative finite-dimensional |${\mathbb{C}}$| -algebra |$A$|⁠ , find algebras |$N$|⁠ , which can be deformed to |$A$|⁠. We develop a simple method that produces associative and flat deformations to investigate this question. As an application of this method we answer a question of Michael Wemyss about deformations of contraction algebras. [ABSTRACT FROM AUTHOR]

This article considers arrowhead and diagonal-plus-rank-one matrices in F n × n where F ∈ { R , C , H } and where H is a noncommutative algebra of quaternions. We provide unified formulas for fast determinants and inverses for considered matrices. The formulas are unified in the sense that the same formula holds in both commutative and noncommutative associative fields or algebras, with noncommutative examples being matrices of quaternions and block matrices. Each formula requires O (n) arithmetic operations, as does multiplication of such matrices with a vector. The formulas are efficiently implemented using the polymorphism or multiple-dispatch feature of the Julia programming language. [ABSTRACT FROM AUTHOR]

Let V be a simple vertex algebra of countable dimension, G be a finite automorphism group of V and σ be a central element of G. Assume that S is a finite set of inequivalent irreducible σ -twisted V -modules such that S is invariant under the action of G. Then there is a finite dimensional semisimple associative algebra A α (G , S) for a suitable 2-cocycle α naturally determined by the G -action on S such that (A α (G , S) , V G) form a dual pair on the sum M of σ -twisted V -modules in S in the sense that (1) the actions of A α (G , S) and V G on M commute, (2) each irreducible A α (G , S) -module appears in M , (3) the multiplicity space of each irreducible A α (G , S) -module is an irreducible V G -module, (4) the multiplicity spaces of different irreducible A α (G , S) -modules are inequivalent V G -modules. As applications, every irreducible V -module is a direct sum of finitely many irreducible V G -modules and irreducible V G -modules appearing in different G -orbits are inequivalent. This result generalizes many previous results. We also establish a bijection between subgroups of G and subalgebras of V containing V G. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an algebra structure denoted by InvDer algebra in which we twist an algebra with an invertible derivation whose inverse is also a derivation. We define InvDer Lie algebras, InvDer associative algebras, InvDer Zinbiel algebras and InvDer dendriforme algebras. We also study the relations between these structures by using Rota–Baxter operators and endomorphism operators. [ABSTRACT FROM AUTHOR]

As a kind of commutative and associative four-dimensional algebra, commutative quaternion has better applications in color image and signal processing than quaternion. Matrix decomposition is of great concern in the theoretical study and numerical calculation of commutative quaternion. Two kinds of disadvantages of commutative quaternion make the decomposition of commutative quaternion matrix extremely difficult. On one hand, commutative quaternion is not a kind of complete four dimensional division algebra because of the zero divisors. On the other hand, computing the inverse of commutative quaternion is very complicated. In this paper, the semi-tensor product (STP) of matrices will be used to overcome the above two kinds of shortcomings. And we will propose a real structure-preserving algorithm based on STP of matrices for commutative quaternion LU decomposition, which makes full use of high-level operations, relation of operations between commutative quaternion matrices and their L-representation matrices. Numerical experiments will be provided to demonstrate the efficiency of the real structure-preserving algorithm based on STP of matrices. Meanwhile, we will apply the structure-preserving algorithm for strict image authentication. [ABSTRACT FROM AUTHOR]

For two given idempotents p and q from an associative algebra A , in this paper, we offer a comprehensive classification of algebras spanned by the idempotents p and q. This classification is based on the condition that p and q are not tightly coupled and satisfy (p q) m − 1 = (p q) m but (p q) m − 2 p ≠ (p q) m − 1 p for some m (≥ 2) ∈ N. Subsequently, we categorize all the group invertible elements and establish an upper bound for the Drazin index of any elements in these algebras spanned by p , q. Moreover, we formulate a new representation for the Drazin inverse of α p + q under two different assumptions, (p q) m − 1 = (p q) m and λ (p q) m − 1 = (p q) m , where α is a non-zero and λ is a non-unit real or complex number. [ABSTRACT FROM AUTHOR]

In this paper, we will solve an important form of hidden discrete logarithm problem (HDLP) and a generalized form of HDLP (GHDLP) over non-commutative associative algebras (FNAAs). We will reduce them to discrete logarithm problem (DLP) in a finite field through analyzing the eigenvalues of the representation matrix. Through the analysis of computational complexity, we will show that HDLP and GHDLP are not good improvements of DLP. With all the instruments in hand, we will break a series of corresponding schemes. Thus, we can conclude that all ideas of constructing cryptographic schemes based on the two solved problems are of no practical significance. [ABSTRACT FROM AUTHOR]

Rota–Baxter operators (RBOs) play a substantial role in many subfields of mathematics, especially in mathematical physics. In the article, RBOs on Zinbiel algebras (ZAs) and their sub-adjacent algebras are first investigated. Moreover, all the RBOs on two and three-dimensional ZAs are presented. Finally, ZAs are also realized in low dimensions of the RBOs of commutative associative algebras. It was found that not all ZAs can be attained in this way. [ABSTRACT FROM AUTHOR]

We investigate general properties of multipliers and weak multipliers of algebras. We apply the results to determine the (weak) multipliers of associative algebras and zeropotent algebras of dimension 3 over an algebraically closed field. [ABSTRACT FROM AUTHOR]

In this paper, we study the finite-dimensional irreducible representations of the BC-graded Lie algebra g 2 n + 1 (ℂ q) by investigating the structure of the BC-graded Lie algebra g 2 n + 1 (R) , where R is a unital involutory associative algebra over a field F of characteristic zero. [ABSTRACT FROM AUTHOR]

We describe the ideals for Mizuhara extensions and find some necessary and sufficient conditions for the simplicity of the direct Mizuhara extension. Also, we study the Mizuhara construction for the matrix algebra and Burde algebras. We construct some various generalizations of the Mizuhara construction and exhibit some examples of the simple pre-Lie algebras that are obtained by this construction; in particular, we construct the simple Witt doubles and for a unital associative commutative algebra with derivation . [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe the irreducible modules for the Jordan algebra of 2 × 2 matrices over an algebraically closed field of characteristic different from 2, 3 and 5 in the class of the commutative power-associative algebras. All irreducible non-unital modules, and irreducible unital modules up to dimension three are classified, namely we find seven non-parameterized and five families of parameterized modules of dimension three. For every k ≥ 2 , an irreducible module of dimension 3k is also constructed. [ABSTRACT FROM AUTHOR]

From the study of the constants in the generic and the degenerate weight subspaces of the free unitary associative complex algebra B, it follows that the constants in the degenerate weight subspaces of the algebra B can be constructed from the corresponding constants in the generic case by a certain specialization procedure. Here we consider that each constant in each generic weight subspace of the algebra B can be expressed by certain iterated q-commutators. [ABSTRACT FROM AUTHOR]

BiHom-associative algebras are generalized associative algebras with two multiplicative linear maps. In this paper, we give the Hochschild homology and cyclic homology structure of BiHom-associative algebras. Then, generalize the dual bimodule action to define the cyclic cohomology. Finally, we introduce the crossed modules of BiHom-associative algebras and show that the Hochschild cohomology of BiHom-associative algebra classifies crossed modules. [ABSTRACT FROM AUTHOR]

The associative Cayley-Dickson algebras over the field of real numbers are also Clifford algebras. The alternative but nonassociative real Cayley-Dickson algebras, notably the octonions and split octonions, share with Clifford algebras an involutary anti-automorphism and a set of mutually anticommutative generators. On the basis of these similarities, we introduce Kingdon algebras : alternative Clifford-like algebras over vector spaces equipped with a symmetric bilinear form. Over three-dimensional vector spaces, our construction quantizes an alternative non-associative analogue of the exterior algebra. The octonions and split octonions, along with other real generalized Cayley-Dickson algebras in Albert's sense, arise as Kingdon algebras. Our construction gives natural characterizations of the octonion and split octonion algebras by a universality property endowing them with a selected superalgebra structure. [ABSTRACT FROM AUTHOR]

The classification of irreducible unital commutative power-associative modules for H n (F) , the algebra of symmetric matrices with the Jordan product, over a field F of characteristic not 2, 3 and 5 are given, for n ≥ 3. It is proved that there exists, up to isomorphisms, only one irreducible module which is not Jordan. It is also shown that every finite dimensional unital commutative power associative module for this algebra is completely reducible. [ABSTRACT FROM AUTHOR]

The representation of elements of free non-associative algebras as a set of multidimensional tables of coefficients is defined. An operation for finding partial derivatives for elements of free non-associative algebras in the same form is considered. Using this representation, a criterion of primitivity for elements of lengths 2 and 3 in terms of matrix ranks, as well as a primitivity test for elements of arbitrary length, is derived. This test makes it possible to estimate the number of primitive elements in free non-associative algebras with two generators over a finite field. The proposed representation allows us to optimize algorithms for symbolic computations with primitive elements. Using these algorithms, we find the number of primitive elements of length 4 in a free non-associative algebra of rank 2 over a finite field. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of local and 2-local derivations of null-filiform, filiform and naturally graded quasi-filiform associative algebras. We prove that these algebras as a rule admit local derivations which are not derivations. We show that filiform and naturally graded quasi-filiform associative algebras admit 2-local derivations which are not derivations and any 2-local derivation of null-filiform associative algebras is a derivation. [ABSTRACT FROM AUTHOR]