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"Real Normed Algebras Revisited," the last paper of the late Gadi Moran, attempts to reconstruct the discovery of the complex numbers, the quaternions, and the octonions, as well as proofs of their properties, using only what was known to 19th-century mathematicians. In his research, Gadi had discovered simple and elegant proofs of the above-mentioned classical results using only basic properties of the geometry of Euclidean spaces and tools from high school geometry. His reconstructions underline an interesting connection between Euclidean geometry and these algebras, and avoid the advanced machinery used in previous derivations of these results. The goal of this article is to present Gadi's derivations in a way that is accessible to a wide audience of readers. [ABSTRACT FROM AUTHOR]

In this note we discuss how a modular triangular origami gift box can be made with six rectangular sheets of printing paper. We also derive a formula for the volume of the box based on high school geometry and algebra. [ABSTRACT FROM AUTHOR]

Let H be a real or complex Hilbert space and $ {\mathcal B}(H) $ B (H) the algebra of all bounded linear operators on H. Recall that a map $ \delta :{\mathcal B}(H)\to {\mathcal B}(H) $ δ : B (H) → B (H) is called an inner Jordan $ * $ ∗ -derivation if there exists some $ T\in \mathcal B(H) $ T ∈ B (H) such that $ \delta (A)=AT-TA^* $ δ (A) = AT − T A ∗ for all $ A\in {\mathcal B}(H) $ A ∈ B (H). In this paper, it is proved that inner Jordan $ * $ ∗ -derivations are the only additive maps δ of $ {\mathcal B}(H) $ B (H) with the property that $ \delta (P)=\delta (P)P^*+P\delta (P) $ δ (P) = δ (P) P ∗ + Pδ (P) for all idempotent operators $ P\in {\mathcal B}(H) $ P ∈ B (H) if $ \dim H=\infty $ dim ⁡ H = ∞ , which is satisfied by additive local Jordan $ * $ ∗ -derivations. For the finite dimensional case, additional conditions are required for δ to be an inner Jordan $ * $ ∗ -derivation. As applications, it is shown that, for any given $ C,D\in {\mathcal B}(H) $ C , D ∈ B (H) , δ satisfies $ \delta (A)B^*+B\delta (A)+\delta (B)A^*+A\delta (B)=D $ δ (A) B ∗ + Bδ (A) + δ (B) A ∗ + Aδ (B) = D for all $ A,B\in {\mathcal B}(H) $ A , B ∈ B (H) with AB + BA = C if and only if δ is an inner Jordan $ * $ ∗ -derivation and $ D=\delta (C) $ D = δ (C). Also, several known results are generalized. [ABSTRACT FROM AUTHOR]

In the present paper, every evolution algebra is endowed with Banach algebra norm. This together with the description of derivations and automorphisms of nilpotent evolution algebras of maximal nilindex, allows the set $ \exp (\mathop {{\rm Der}}(\mathbf{E})) $ exp ⁡ (Der ⁡ (E)) to be investigated. Moreover $ \exp (\mathop {{\rm Der}}({\mathbf{E}})) $ exp ⁡ (Der ⁡ (E)) is shown to be a normal subgroup of $ \mbox {Aut}(\mathbf{E}) $ Aut (E) , and its corresponding index is calculated. [ABSTRACT FROM AUTHOR]

In the present paper corrected versions of the statements in the paper "Description of finite-dimensional inner Rickart and Baer Jordan algebras" by F.N. Arzikulov and U.I. Khakimov are given. In particular, it is shown that for any Jordan algebra J with an idempotent p and an associative degenerate radical D such that J = F p + ̇ D , J is an inner RJ-algebra if and only if, for any nonzero a ∈ D , a 2 = 0 and p(pa) = pa. Also, other equivalent conditions when a Jordan algebra J is an inner RJ-algebra are given. As for finite-dimensional nilpotent Jordan algebras, there is not a nilpotent inner RJ-algebra (and hence inner BJ-algebra) except the finite-dimensional Jordan algebra the square of each element of which is zero. [ABSTRACT FROM AUTHOR]

In a previous paper we have compared the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two idempotents. In the present paper, we compare the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two arrows. [ABSTRACT FROM AUTHOR]

This paper is devoted to the descriptions of automorphisms and local automorphisms on complex solvable Lie algebras of maximal rank. First, it is established that any automorphism on a solvable Lie algebra of maximal rank can be represented as a product (composition) of inner, diagonal and graph automorphisms. We apply the description of automorphism to the specification of automorphisms on solvable Lie algebras of maximal rank with abelian nilradical, and to the description of automorphisms of standard Borel subalgebras of complex simple Lie algebras. Based on the representation of an automorphism, it is proved that all local automorphisms on a solvable Lie algebra of maximal rank are global automorphisms. We also present two examples of solvable Lie algebras which are not of maximal rank and have different behaviours of local automorphisms. Namely, the first algebra does not admit pure local automorphisms, while the second algebra admits a local automorphism which is not an automorphism. [ABSTRACT FROM AUTHOR]

In Fialowski's classification for algebras of maximal class, there are three Lie algebras of maximal class with one-dimensional homogeneous components: $ \mathfrak {m}_0 $ m 0 , $ L_1 $ L 1 and $ \mathfrak {m}_2 $ m 2 . In this paper, we study their biderivations by considering the embedded mapping to derivation algebras. Then we determine commuting mappings on these algebras as an application of biderivations. Finally, local and 2-local derivations for these three algebras are characterized as the given gradings. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of marking constraints in a class of partially controllable timed Petri nets. Specifically, these are Timed Event Graphs (TEGs) which include disturbed input transitions and which are subject to token capacity constraints in some paths. Using the Min-Plus algebra formalism, formal approaches are proposed to design causal control laws to guarantee compliance with marking constraints in TEGs. Sufficient conditions are defined to show the existence and synthesis of these controllers. The developed methodologies are illustrated in a case study of a disassembly system with a limited stock of components. [ABSTRACT FROM AUTHOR]

Let $ n \geq ~2 $ n ≥ 2 be a natural number, and denote by $ \mathcal {M}_{n} $ M n the algebra of all $ n \times n $ n × n matrices over an algebraically closed field $ \mathbb {F} $ F of zero characteristic. Let also $ K_{1} $ K 1 and $ K_{2} $ K 2 be two non-empty proper subsets of $ \mathbb {F} $ F . In this paper, we characterize linear maps φ on $ \mathcal {M}_{n} $ M n having the property that, for every $ T \in \mathcal {M}_{n} $ T ∈ M n , the spectrum of T is a subset of $ K_1 $ K 1 if and only if the spectrum of $ \varphi (T) $ φ (T) is a subset of $ K_2 $ K 2 . We obtain a similar caracterization for the case when $ K_{1} $ K 1 and $ K_{2} $ K 2 have both at most n elements, working with the equality of the spectrum to the fixed subsets instead of the inclusion. [ABSTRACT FROM AUTHOR]

In this paper, we present a modality-free pre-rough algebra. Łukasiewicz Moisil algebra and Wajsberg algebra are equivalent under a transformation. A similar type of equivalence exists in our proposed definition and standard definition of pre-rough algebra. We obtain a few modality-free algebras weaker than pre-rough algebra. Furthermore, it is also established that modality-free versions for other analogous structures weaker than pre-rough algebra do not exist. Both Hilbert-type axiomatization and sequent calculi for all proposed algebras are presented. [ABSTRACT FROM AUTHOR]

Over the past 50 years, Nelson algebras have been extensively studied by distinguished scholars as the algebraic counterpart of Nelson's constructive logic with strong negation. Despite these studies, a comprehensive survey of the topic is currently lacking, and the theory of Nelson algebras remains largely unknown to most logicians. This paper aims to fill this gap by focussing on the essential developments in the field over the past two decades. Additionally, we explore generalisations of Nelson algebras, such as N4-lattices which correspond to the paraconsistent version of Nelson's logic, as well as their applications to other areas of interest to logicians, such as duality and rough set theory. A general representation theorem states that each Nelson algebra is isomorphic to a subalgebra of a rough set-based Nelson algebra induced by a quasiorder. Furthermore, a formula is a theorem of Nelson logic if and only if it is valid in every finite Nelson algebra induced by a quasiorder. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of 2-local derivations on the Schrödinger-Virasoro Algebra, which is an infinite-dimensional Lie algebra with three outer derivations. We prove that all 2-local derivations on the Schrödinger-Virasoro Algebra are derivations. [ABSTRACT FROM AUTHOR]

In this paper, we give a complete classification of all free $ U(\mathbb {C}L_0 \oplus \mathbb {C}Y_0\oplus \mathbb {C}M_0) $ U (C L 0 ⊕ C Y 0 ⊕ C M 0) -modules of rank 1 over a Schrödinger-Virasoro type algebra $ \mathfrak {tsv} $ tsv . [ABSTRACT FROM AUTHOR]

Fostering students' relational understanding of the equals sign is a challenge for math educators that begins in the primary levels and persists into tertiary education. In this paper, I develop an entry point, especially for students who only have an operational understanding of the equals sign, to the core idea of equivalence in linear algebra. My approach is informed by the history of mathematics: In the 17th and 18th centuries, mathematics research underwent an algebraicization, with mathematicians replacing their classical geometric questions with novel algebraic investigations. In this paper, I will offer geometric interpretations of two operations developed at the precipice of this monumental shift: algebraic substitution and Gaussian elimination. I will then utilize Lakoff & Johnson's theory of conceptual metaphor to compare and contrast this historically-grounded geometric re-interpretation of modern linear algebra to the direct algebraic interpretation taken in most modern textbooks. [ABSTRACT FROM AUTHOR]

In this note we describe the mathematics that emerges from the construction of an origami box. We first construct a simple origami box from two rectangular sheets and then discuss some of the mathematical questions that arise in the context of algebra, geometry and calculus. [ABSTRACT FROM AUTHOR]

The present paper is devoted to studying local derivations on the Lie algebra $ W(2,2) $ W (2 , 2) which has some outer derivations. Using some linear algebra methods in [1] and a key construction for $ W(2,2) $ W (2 , 2) , we prove that every local derivation on $ W(2, 2) $ W (2 , 2) is a derivation. As an application, we determine all local derivations on the deformed $ \mathfrak {bms}_3 $ b m s 3 algebra. [ABSTRACT FROM AUTHOR]

In this work, we study the relational representation of the class of Tarski algebras endowed with a subordination, called subordination Tarski algebras. These structures were introduced in a previous paper as a generalisation of subordination Boolean algebras. We define the subordination Tarski spaces as topological spaces with a fixed basis endowed with a closed relation. We prove that there exist categorical dualities between categories whose objects are subordination Tarski algebras and categories whose objects are subordination Tarski spaces. These results extend the known dualities for modal algebras. Finally, we are going to characterise some algebraic conditions written in the quasi-modal language by means of first-order conditions. [ABSTRACT FROM AUTHOR]

Two groups of mathematics faculty, one from a four-year college and one from a two-year college, redesigned their respective introductory college mathematics courses following presentation of observational data regarding how faculty had been teaching the courses. This presentation emphasised how infrequently faculty teaching introductory college mathematics employed recommended pedagogical practices. This work was part of a multi-year federal grant project designed to increase the numbers of underrepresented students majoring in STEM disciplines. While the two teams developed very different redesigned course activities, in both instances the primary motivation for initiating the work was the information provided to faculty in a professional development workshop regarding how they had previously been observed teaching the mathematics content and how infrequently they utilised the pedagogical practices recommended by STEM education experts. The paper also highlights faculty resistance to curricular reform and enumerates some ways of addressing this resistance. The paper also discusses why faculty from two-year and four-year institutions resisted working together on course redesign and provides recommendations for future efforts addressing course redesign. [ABSTRACT FROM AUTHOR]

Quasi-Nelson logic (QNL) was recently introduced as a common generalisation of intuitionistic logic and Nelson's constructive logic with strong negation. Viewed as a substructural logic, QNL is the axiomatic extension of the Full Lambek Calculus with Exchange and Weakening by the Nelson axiom, and its algebraic counterpart is a variety of residuated lattices called quasi-Nelson algebras. Nelson's logic, in turn, may be obtained as the axiomatic extension of QNL by the double negation (or involutivity) axiom, and intuitionistic logic as the extension of QNL by the contraction axiom. A recent series of papers by the author and collaborators initiated the study of fragments of QNL, which correspond to subreducts of quasi-Nelson algebras. In the present paper we focus on fragments that contain the connectives forming a residuated pair (the monoid conjunction and the so-called strong Nelson implication), these being the most interesting ones from a substructural logic perspective. We provide quasi-equational (whenever possible, equational) axiomatisations for the corresponding classes of algebras, obtain twist representations for them, study their congruence properties and take a look at a few notable subvarieties. Our results specialise to the involutive case, yielding characterisations of the corresponding fragments of Nelson's logic and their algebraic counterparts. [ABSTRACT FROM AUTHOR]

This work is devoted to the complete classification of solvable Leibniz algebras with the naturally graded quasi-filiform non-Lie nilradical. It should be noted that the class of naturally graded quasi-filiform Leibniz algebras is sufficiently large, and solvable Leibniz algebras with the nilradical which is isomorphic to several of these algebras are classified in a series of works. This paper considered all omitted cases and completed the problem of classifying solvable Leibniz algebras with naturally graded quasi-filiform non-Lie nilradical. At the end of the paper we combine all results obtained to date, and provide a complete list of solvable Leibniz algebras, whose nilradical is a naturally graded quasi-filiform Leibniz algebra. [ABSTRACT FROM AUTHOR]

The generalized L'vov–Kaplansky conjecture states that for any finite-dimensional simple algebra A the image of a multilinear polynomial on A is a vector space. In this paper, we prove it for the algebra of octonions $ \mathbb {O} $ O over a field F satisfying certain specified conditions (in particular, we prove it for quadratically closed fields, and for the field $ \mathbb R $ R ). In fact, letting V be the space of pure octonions in $ \mathbb {O} $ O , we prove that the image set must be either $ \{0\} $ { 0 } , F, V or $ \mathbb {O} $ O . We discuss possible evaluations of semihomogeneous polynomials on $ \mathbb {O} $ O and of arbitrary polynomials on the corresponding Malcev algebra. [ABSTRACT FROM AUTHOR]

Köthe's classical problem posed by G. Köthe in 1935 asks to describe the rings R such that every left R-module is a direct sum of cyclic modules (these rings are known as left Köthe rings). Köthe, Cohen and Kaplansky solved this problem for all commutative rings (that are Artinian principal ideal rings). During the years 1962 to 1965, Kawada solved Köthe's problem for basic finite-dimensional algebras. But, so far, Köthe's problem was open in the non-commutative setting. Recently, in the paper [Several characterizations of left Köthe rings, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2023) (to appear)]], we classified left Köthe rings into three classes one contained in the other: left Köthe rings, strongly left Köthe rings and very strongly left Köthe rings, and then, we solved Köthe's problem by giving several characterizations of these rings in terms of describing the indecomposable modules. In this paper, we will introduce the Morita duals of these notions as left co-Köthe rings, strongly left co-Köthe rings and very strongly left co-Köthe rings, and then, we give several structural characterizations for each of them. [ABSTRACT FROM AUTHOR]

The notion of the weighted core inverse in a ring with involution was introduced recently [Mosić et al. Comm. Algebra, 2018; 46(6); 2332–2345]. In this paper, we explore new characterizations of the weighted core inverse of sum and the difference between two weighted core invertible elements in a ring with involution under different conditions. Further, we discuss reverse order and mixed-type reverse order laws for the weighted core invertible elements in a ring. [ABSTRACT FROM AUTHOR]

Recently, relative Rota–Baxter (Lie/associative) algebras are extensively studied in the literature from cohomological points of view. In this paper, we consider relative Rota–Baxter Leibniz algebras (rRB Leibniz algebras) as the object of our study. We construct an $ L_\infty $ L ∞ -algebra that characterizes rRB Leibniz algebras as its Maurer–Cartan elements. Then we define representations of an rRB Leibniz algebra and introduce cohomology with coefficients in a representation. As applications of cohomology, we study deformations and abelian extensions of rRB Leibniz algebras. [ABSTRACT FROM AUTHOR]

In this paper, we construct a class of non-weight modules over the affine-Virasoro algebra of type A1 by taking the tensor product of irreducible U(h)-free modules of rank 1 with irreducible highest weight modules. The irreducibility and the isomorphism classes of these modules are determined. Moreover, we show that these tensor product modules are different from the known non-weight modules. Finally, we realize some tensor product modules as induced modules from modules over certain subalgebras of the affine-Virasoro algebra of type A1, and give sufficient and necessary conditions for these induced modules to be reducible. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the relationships between a Hom-Jordan algebras and its induced ternary Hom-Jordan algebras. We give some properties of the αk-generalized derivation algebra G D e r (J) of a ternary Hom-Jordan algebras. In particular, we prove that G D e r (J) = Q D e r (J) + Q Γ (J) , the sum of the αk-quasiderivation algebra and the αk-quasicentroid. We also prove that Q D e r (J) can be embedded as derivations in a larger ternary Hom-Jordan algebras. General results on αk-centroids of ternary Hom-Jordan algebras are also developed in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the structure of the variety of Lie algebras with a finite number of ideals and their graph representations using Hasse diagrams. The large number of necessary conditions on the algebraic structure of this type of algebras leads to the explicit description of those algebras in the variety with trivial Frattini subalgebra. To illustrate our results, we have included and discussed lots of examples throughout the paper. [ABSTRACT FROM AUTHOR]

AbstractInversion sequences of length n are positive integer sequences e1 e2 …en such that 1 ≤ ei ≤ i for all 1 ≤ i ≤ n. These sequences are in bijection with the permutations of [n]. This paper focuses on the polyomino or barpgraph representation of the inversion sequences. More precisely, we study the distribution of lattice points on these polyominoes. We find the generating functions respect to the length, the number of interior vertices, corners, and vertices of a given degree. We also give simple explicit formulas for the total values of these parameters over all polyominoes of inversion sequences of length n. Throughout this work, we use symbolic computer algebra to facilitate the computations. [ABSTRACT FROM AUTHOR]

Let (X , ≤) be a locally finite pre-ordered set and R be a commutative ring with unity. In this paper we apply the theory of zero product determined algebras to show that each linear map on the incidence algebra I (X , R) which is derivable at zero is a generalized derivation and every local derivation on I (X , R) is a derivation. [ABSTRACT FROM AUTHOR]

Since the existing tropical cryptographic protocols are either susceptible to the Kotov-Ushakov attack and its generalization, or to attacks based on tropical matrix periodicity and predictive behavior, several attempts have been made to propose protocols that resist such attacks. Despite these attempts, many of the proposed protocols remain vulnerable to attacks targeting the underlying hidden problems, one of which we call the tropical two-sided discrete logarithm with shift. An illustrative case is the tropical Stickel protocol, which, when formulated with a single monomial instead of a polynomial, becomes susceptible to attacks based on solutions of the above mentioned tropical version of discrete logarithm. In this paper we will formally introduce the tropical two-sided discrete logarithm with shift, discuss how it is solved, and subsequently demonstrate an attack on a key exchange protocol based on the tropical semiring of pairs. This particular protocol is compromised due to the existence of efficient (albeit heuristic) solution of the tropical two-sided logarithm problem, and this highlights the ongoing challenges in search of a "good" key exchange protocol in tropical cryptography. [ABSTRACT FROM AUTHOR]

In this paper, Whittaker modules are studied for a subalgebra q ϵ of the N =2 superconformal algebra. The Whittaker modules are classified by central characters. Additionally, criteria for the irreducibility of the Whittaker modules are given. [ABSTRACT FROM AUTHOR]

Let A be an associative superalgebra endowed with a pseudoautomorphism p. In this paper we generalize the Wedderburn-Malcev Theorem in this setting and we prove that the sequence of p-codimensions of A is polynomially bounded if and only if the variety generated by A does not contain the group algebra of Z 2 and the algebra of 2 × 2 upper-triangular matrices with suitable pseudoautomorphisms. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, all biderivations of solvable Lie algebras of maximal rank g=Q⊕N are characterized. Namely, we consider a solvable Lie algebra of the form g=Q⊕N , where Q is the maximal torus subalgebra of g, N is the nilradical of g and dimQ=dimN/N2 .In the case dimQ=N we characterize the form of skew-symmetric and symmetric biderivations of g . As applications, the forms of linear commuting (skew-commuting) maps and the commutative post-Lie algebra structures on g are given. [ABSTRACT FROM AUTHOR]

AbstractThis paper starts by showing that, for algebras in a certain class, the concepts of weak nilpotency and nilpotency coincide. It goes on to describe some solvability and nilpotency properties of bicommutative algebras, of assosymmetric algebras and of Novikov algebras and to introduce a Frattini theory for all of them. A description is also given for semisimple bicommutative algebras over any field. [ABSTRACT FROM AUTHOR]

AbstractLet k be an arbitrary field and let b>1,n>1 and a be three positive integers. In this paper we explicitly describe a minimal S-graded free resolution of the semigroup algebra k[S] when S is a generalized repunit numerical semigroup, that is, when S is the submonoid of N generated by {a1,a2,…,an} where a1=∑j=0n−1bj and ai−ai−1=a bi−2, i=2,…,n, with gcd(a,a1)=1. [ABSTRACT FROM AUTHOR]

In this paper we studied the personal mathematical work of 14 students in initial teacher training, in their first year of studies at a public university in Chile, based on two tasks on complex numbers. These tasks were set in a Computer Aided Assessment System (CAA) based on Moodle and Wiris and it was proposed to solve them using different Computer Algebra Systems (CAS), such as GeoGebra, Symbolab, Photomath and Wolfram Alpha among others. Difficulties and potentialities were observed in the mathematical work of the students. In the difficulties, it was observed that the students had problems interpreting the information from the CAS and the CAA feedback because their previous mathematical knowledge was not solid enough to do so. In the potentialities, it was observed that different characteristics of the task, together with the articulation of two or more artifacts allowed students to give meaning to the mathematical objects involved. [ABSTRACT FROM AUTHOR]

In this paper, we first investigate some properties of relative Rota-Baxter operators on Leibniz conformal algebras with respect to representations and their connections with Leibniz dendriform conformal algebras. Next, we introduce the notion of a twisted relative Rota-Baxter operator and construct a conformal NS-Leibniz algebra structure related to twisted relative Rota-Baxter operators. Furthermore, we define the cohomology of a twisted relative Rota-Baxter operator with coefficients in a suitable representation. Finally, we consider formal deformations of twisted relative Rota-Baxter operators from cohomological points of view. [ABSTRACT FROM AUTHOR]

Let D be an F-central division algebra. In this paper, we investigated the exponent of the group G (D) = D * / Nrd D (D *) D ′ , where D * is the group of units of D, Nrd D (D *) is the image of D * under the reduced norm map and D ′ is the commutator subgroup of D * . We show that if exp (G (D)) < ind (D) , then D and F satisfy strong conditions. In particular, we observe that if D is a sum cyclic algebras in Br (F) , then exp (G (D)) < ind (D) if and only if F is euclidean and D is a tensor product of an ordinary quaternion algebra and a division algebra of odd index. [ABSTRACT FROM AUTHOR]

In this paper, we show that higher Hochschild homology groups do not vanish for a class of Hochschild extension algebras of truncated quiver algebras by the standard duality module. Consequently, this result extends the result of the trivial extension algebra by the standard duality module for truncated quiver algebras. Moreover, as an application, we show that for any Hochschild extension algebra of selfinjective Nakayama algebras by the standard duality module, its Hochschild homology dimension is infinite. [ABSTRACT FROM AUTHOR]

In this paper we study the isotopism classes of two-step nilpotent algebras. We show that every nilpotent Leibniz algebra g with dim [ g , g ] = 1 is isotopic to the Heisenberg Lie algebra or to the Heisenberg algebra l 2 n + 1 J 1 , where J1 is the n × n Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with one-dimensional commutator ideal. Eventually, we introduce new isotopism invariants for Leibniz algebras and Lie racks. [ABSTRACT FROM AUTHOR]

Motivated by recent work on compatible mock-Lie algebras such that any linear combination of the two mock-Lie structures is a mock-Lie structure, in this paper we introduce compatible mock-Lie bialgebras as an analogue of compatible Lie bialgebras. They can also be regarded as a "compatible version" of mock-Lie bialgebras. Constructions of compatible mock-Lie bialgebras are presented via Manin triples and matched pairs. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we use the Balinsky-Novikov construction to naturally realize the super Heisenberg-Virasoro algebra g . We then proceed to compute the second cohomology of g with trivial coefficients and determine its universal central extension. This yields two odd 2-cocycles in addition to the Virasoro 2-cocycle. We also calculate the first cohomology group of g in the adjoint module to classify all derivations of g . Finally, we classify all automorphisms of g and describe its automorphism group. [ABSTRACT FROM AUTHOR]

AbstractThis paper is concerned with generalizing the results for Lie CT-algebras to Leibniz algebras. In some cases our results give a generalization even for the case of a Lie algebra. Results on A-algebras are used to show every Leibniz CT-algebra over an algebraically closed field of characteristic different from 2,3 is solvable or is isomorphic to sl2(F). A characterization is then given for solvable Leibniz CT-algebras. It is also shown that the class of solvable Leibniz CT-algebras is factor closed. [ABSTRACT FROM AUTHOR]

AbstractLeibniz algebras are certain generalizations of Lie algebras. Motivated by the concept of subinvariance and ascendancy in group theory, Schenkman studied properties of subinvariant subalgebras of a Lie algebra. Kawamoto studied ascendency in Lie algebras. In this paper we define subinvariance and asendency in Leibniz algebras and study their properties. It is shown that the signature results on subinvariance and ascendency in Lie algebras have analogs for Leibniz algebras. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we introduce a combinatorial path model of representation of the quantum affine algebra of type Dn, inspired by Mukhin and Young’s combinatorial path models of representations of the quantum affine algebras of types An and Bn. In particular, we give a combinatorial formula for q-characters of fundamental modules of type Dn by assigning each path to a monomial or binomial. By counting our paths, a new expression on dimensions of fundamental modules of type Dn is obtained. [ABSTRACT FROM AUTHOR]

Making connections within and between different aspects of mathematics is recognised as fundamental to learning mathematics with understanding. However, exactly what these connections are and how they serve the goal of learning mathematics is rarely made explicit in curriculum documents with the result that mathematics tends to be presented as a set of discrete, disconnected topics. Interest in establishing a more coherent approach to the teaching and learning of school mathematics has led to a focus on big ideas. That is, networks of related concepts, skills and ways of thinking that facilitate learning mathematics with understanding. Research on learning progressions has helped identify what these big ideas are and how they serve to build connections within and between different aspects of mathematics. This paper draws on research that provides an evidenced-based learning progression for multiplicative reasoning to illustrate the connective role of multiplicative thinking in the development of algebraic, geometrical, and statistical reasoning. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we extend the notion of anti-pre-Lie algebras to the ℤ2-graded version, and introduce the notion of anti-pre-Lie superalgebras. They can be characterized as a class of Lie-admissible superalgebras that satisfy its negative left multiplication operators are the representations of the corresponding sub-adjacent Lie superalgebras. And we give the classification of 2-dimensional anti-pre-Lie superalgebras. We introduce the notion of anti-super -operators on Lie superalgebras to explore the relationships between anti-super -operators and anti-pre-Lie superalgebras. We show that nondegenerate super-commutative 2-cocycles on Lie superalgebras can obtain a class of compatible anti-pre-Lie superalgebra structures. In addition, we introduce a subclass of anti-pre-Lie superalgebras, namely admissible Novikov superalgebras, which correspond to Novikov superalgebras through q-superalgebras. Finally, we introduce the notions of anti-pre-Lie Poisson superalgebras and admissible Novikov-Poisson superalgebras, extending the correspondence to the level of Poisson type structures, the correspondence of the Novikov-Poisson superalgebras and the admissible Novikov-Poisson superalgebras are realized through admissible pairs. [ABSTRACT FROM AUTHOR]

In this paper, first we give the controlling algebra of Lie triple systems. In particular, the cohomology of Lie triple systems can be characterized by the controlling algebra. Then using controlling algebras, we introduce the notions of homotopy Nambu algebras and homotopy Lie triple systems. We show that 2-term homotopy Lie triple systems is equivalent to Lie triple 2-systems, and the latter is the categorification of a Lie triple system. Finally we study skeletal and strict Lie triple 2-systems. We show that skeletal Lie triple 2-systems can be classified the third cohomology group, and strict Lie triple 2-systems are equivalent to crossed modules of Lie triple systems. [ABSTRACT FROM AUTHOR]

The notion of pre-Leibniz algebras was recently introduced in the study of Rota-Baxter operators on Leibniz algebras. In this paper, we first construct a graded Lie algebra whose Maurer-Cartan elements are pre-Leibniz algebras. Using this characterization, we define the cohomology of a pre-Leibniz algebra with coefficients in a representation. This cohomology is shown to split the Loday-Pirashvili cohomology of Leibniz algebras. As applications of our cohomology, we study formal and finite order deformations of a pre-Leibniz algebra. Finally, we define homotopy pre-Leibniz algebras and classify some special types of homotopy pre-Leibniz algebras. [ABSTRACT FROM AUTHOR]