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For a commutative ring R and a Hopf algebra H which is finitely generated projective as an R-module, it is established that there is an (anti)-isomorphism of groups between the Brauer group BQ(R, H) of Hopf Yetter-Drinfel'd H-module algebras and the Brauer group BQ (R , H *) of Hopf Yetter-Drinfel'd H * -module algebras, where H * is the linear dual of H. In this paper, we generalize this result by establishing an anti-isomorphism of groups between BQ(S, H), the Brauer group of dyslectic Hopf Yetter-Drinfel'd (S, H)-module algebras and BQ (S o p , H *) , the Brauer group of dyslectic Hopf Yetter-Drinfel'd (S o p , H *) -module algebras, where S is an H-commutative Hopf Yetter-Drinfel'd H-module algebra and Sop is the opposite algebra of S. Communicated by Alberto Elduque [ABSTRACT FROM AUTHOR]

This paper aims to study the local derivations, 2-local automorphisms and local automorphisms on the super-Virasoro algebras. The primary focus is to establish that every local derivation of the super-Virasoro algebras is indeed a derivation, and to demonstrate that every local or 2-local automorphism of the super-Virasoro algebras is an automorphism. [ABSTRACT FROM AUTHOR]

The present paper is devoted to study local derivations on the conformal Galilei algebra. We prove that every local derivation on the conformal Galilei algebra is a derivation. [ABSTRACT FROM AUTHOR]

Differential operators and integral operators are linked together by the first fundamental theorem of calculus. Based on this principle, the notion of a differential Rota-Baxter algebra was proposed by Guo and Keigher. Recently, the subject has attracted more attention since it is associated with many areas in mathematics, such as integro-differential algebras. This paper considers differential Rota-Baxter algebras in the quasi-idempotent operator context. We establish a Gröbner-Shirshov basis for free commutative quasi-idempotent differential algebras (resp. Rota-Baxter algebras, resp. differential Rota-Baxter algebras). This provides a linear basis of a free object in each of the three corresponding categories by the Composition-Diamond lemma. Communicated by P. Kolesnikov [ABSTRACT FROM AUTHOR]

In the paper, we show that a uniformly bounded and nonnegative triangle functor between Gorenstein derived categories of two CM-algebras induces a Gorenstein-projectively stable functor between the corresponding Gorenstein-projectively stable categories. Furthermore, we show that a ladder of Gorenstein derived categories of CM-finite algebras induces a recollement of Gorenstein-projectively stable categories under certain nice conditions. As a byproduct, a Gorenstein derived equivalence induces a Gorenstein-projectively stable equivalence for CM-finite Gorenstein algebras. [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]

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]

We view the (real) Laplace transform through the lens of linear algebra as a continuous analogue of the power series by a negative exponential transformation that switches the basis of power functions to the basis of exponential functions. This approach immediately points to how the complex Laplace transform is a generalisation of the Fourier transform where the pole of the transform realises the linear algebraic intuition. The exponential transformation also motivates the Taylor inversion of the real Laplace transform. [ABSTRACT FROM AUTHOR]

Let R be a ring and S be a multiplicative subset of R. Then R is called a uniformly S-Noetherian ring if there exists s ∈ S such that, for any ideal I of R, sI ⊆ K for some finitely generated subideal K of I. We give the Eakin-Nagata-Formanek theorem for uniformly S-Noetherian rings. In addition, the uniformly S-Noetherian properties on several ring constructions are given. The notion of u-S-injective modules is also introduced and studied. Finally, we obtain the Bass-Papp theorem for uniformly S-Noetherian rings. [ABSTRACT FROM AUTHOR]

In the paper, we describe the automorphisms on the solvable Leibniz algebras with the model or abelian nilradicals, whose complementary spaces are equal to two and one, respectively. We show that any local automorphism on a solvable Leibniz algebra with model nilradical of maximal codimension is an automorphism. We show that solvable Leibniz algebras with abelian nilradicals, which have 1-dimension complementary space, admit local automorphisms which are not automorphisms. Moreover, similar problems concerning 2-local automorphisms of such algebras are investigated. We give examples of solvable Leibniz algebras in which 2-local automorphisms are automorphisms and examples of such algebras which admit 2-local automorphisms which are not automorphisms. [ABSTRACT FROM AUTHOR]

In this paper, the super-skewsymmetric super-biderivations on the N = 1 super Heisenberg-Virasoro algebra are completely determined. Furthermore, we find that every super-skewsymmetric super-biderivation of the N = 1 super Heisenberg-Virasoro algebra is inner. [ABSTRACT FROM AUTHOR]

Assume that B is a finite dimensional algebra, and A = B [ P 0 ] is the one-point extension algebra of B using a finitely generated projective B-module P0. The categories of B-modules and A-modules are connected via two adjoint functors known as the restriction and extension functors, denoted by R and E , respectively. These functors have nice homological properties and have been studied in the category mod- A of finitely presented modules that we extend to the category Mod- A of all A-modules. Our main focus is to investigate the behavior of important subcategories (tilting and τ-tilting subcategories) and objects (finendo quasi-tilting modules, silting modules, and cosilting modules) under these functors. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give some constructions results and examples of n-BiHom-Lie color algebras. Next, we introduce the definition of BiHom-modules over n-BiHom-Lie color algebras and we provide some properties. Moreover we investigate generalized derivations of n-BiHom-Lie color algebras and their BiHom-subalgebras. [ABSTRACT FROM AUTHOR]

Let p and q, where p q − q p = 1 , be the standard generators of the first Weyl algebra A1 over a field of characteristic zero. Then the spectrum of the inner derivation ad(pq) on A1 are exactly the set of integers. The algebra A1 is a Z -graded algebra with each i-component being the i-eigenspace of ad(pq), where i ∈ Z . Assume that z and w are elements of A1 satisfying z w − w z = 1 . The Dixmier Conjecture for A1 says that they always generate A1. We show that if z possesses no component belonging to the negative spectrum of ad(pq), then z and w generate A1. We give some generalization of this result, and some other useful criterions for z and w to generate A1. It is shown that if z is a sum of not more than 2 homogeneous elements of A1 then z and w generate A1, which generalizes a known result due to Bavula and Levandovskyy. [ABSTRACT FROM AUTHOR]

Let K be a field, V a finite dimensional K-vector space and E the exterior algebra of V. We analyze iterated mapping cone over E. If I is a monomial ideal of E with linear quotients, we show that the mapping cone construction yields a minimal graded free resolution F of I via the Cartan complex. Moreover, we provide an explicit description of the differentials in F when the ideal I has a regular decomposition function. Finally, we get a formula for the graded Betti numbers of a new class of monomial ideals including the class of strongly stable ideals. [ABSTRACT FROM AUTHOR]

In this paper, we provide a new approach to construct monoidal Hom-Hopf algebras. We investigate monoidal Hom-Hopf algebra structure on a left (n, l)-Hom-crossed coproduct structure with a left n-Hom-smash product structure, obtaining Radford [ n , (n , l) ] -biproduct structure theorem. Then, we study a Hom-coaction admissible mapping system to characterize this Radford [ n , (n , l) ] -biproduct structure. Finally, we study the cosemisimplicity of a special Hom-smash coproduct and prove the related Maschke theorem. [ABSTRACT FROM AUTHOR]

Let A be a non-local self-injective algebra of polynomial growth over an algebraically closed field. We show that the set of simple modules is not contained in a quasi-tube. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let A be a G-crossed product O -algebra where O is a commutative neotherian local complete ring such that the field k = O / J (O) is of characteristic p. We study the Clifford Theory of A over the 1-component A1 of A with emphasis on the vertex-source pairs of indecomposable A-modules. We present the following application of our results (Section 5): Let O = k be a field of characteristic p that is "big enough", let b ∈ B l (A) so that b ∈ A 1 , let e ∈ B l (A 1) that is covered by b and let T be the G-stabilizer of e. Thus e is a central idempotent of AT. As is known, there is a twisted group algebra C of T over k such that the categories C-mod and Ab-mod are equivalent. Let X be an indecomposable (resp. simple) C-module. Then from a vertex-source pair of X, we give a formula for a vertex-source pair of a corresponding indecomposable (resp. simple) Ab-module X * . [ABSTRACT FROM AUTHOR]

As a double generalization of biFrobenius algebras and weak Hopf algebra, we introduce and study weak biFrobenius algebras (or briefly wbF algebras). A wbF algebra is a Frobenius algebra and a Frobenius coalgebra and satisfies some compatibility conditions. We discuss more properties of wbF algebras keeping as close as possible to the classical theory of weak Hopf algebras, and give conditions for finite dimensional algebras and coalgebras to be wbF algebras. Then we studies substructures in wbF algebras. Finally, we prove that our wbF algebras can give rise to Frobenius double algebras in sense of Szlachányi. [ABSTRACT FROM AUTHOR]

Given a skew-symmetric matrix X, the Pfaffian of X is defined as the square root of the determinant of X. In this article, we give the explicit defining equations of the Rees algebra of a Pfaffian ideal I generated by the maximal order Pfaffians of a generic skew-symmetric matrix. We further prove that all diagonal subalgebras of the corresponding Rees algebra of I are Koszul. We also look at Rees algebras of Pfaffian ideals of linear type associated with certain sparse skew-symmetric matrices. In particular, we consider the tridiagonal matrices and identify the corresponding Pfaffian ideals to be of Gröbner linear type and as the vertex cover ideals of unmixed bipartite graphs. As an application of our results, we conclude that all their ordinary and symbolic powers have linear quotients. [ABSTRACT FROM AUTHOR]

In this paper, we study the corona theorem and Wolff theorem for the multiplier algebra of the Besov–Morrey space $ \mathcal {B}_p^{\lambda }(s) $ B p λ (s) when $ 0 0 < λ , s < 1 < p < ∞. Here the Besov–Morrey space $ \mathcal {B}_p^{\lambda }(s) $ B p λ (s) is the space consisting of those analytic functions on the unit disc $ {\mathbb D} $ D such that $ |f'(z)|^p(1-|z|^2)^{p+s-2}{\rm d}A(z) $ | f ′ (z) | p (1 − | z | 2) p + s − 2 d A (z) is a $ \lambda s $ λs -Carleson measure. [ABSTRACT FROM AUTHOR]

We describe how an instructor integrated embodiment to teach the Fundamental Homomorphism Theorem (FHT) and preliminary concepts in an undergraduate abstract algebra course. The instructor's use of embodiment reduced levels of abstraction for formal definitions, theorems, and proofs. The instructor's simultaneous use of various forms of embodiment primed students for the formalism and symbolism, highlighted and disambiguated students' referents, amplified students' contributions to develop conceptual fluency, and linked students' body form catchments to motivate the FHT. Our results offer practical implications for teaching by illustrating examples of how embodiment can assist in making abstract concepts concrete. [ABSTRACT FROM AUTHOR]

South African learners' performance in mathematics, both locally and internationally, has raised significant concerns, particularly in the realm of algebra. To address this issue, a mixed-methods study was conducted to explore the effectiveness of using algebra tiles to teach algebraic expressions. The study aimed to investigate algebraic expression errors and misconceptions among Grade 9 learners and evaluate the impact of an intervention involving algebra tiles on learners' post-test results. Data were collected through tests administered to a class of 22 Grade 9 learners. The findings of the study confirmed the error types identified in the literature and demonstrated a notable improvement in performance on the post-test following the intervention using algebra tiles. The results indicated that the intervention successfully rectified pre-existing errors and misconceptions, resulting in an 18% enhancement in overall performance among participants. The study aligned with a Vygotskian sociocultural perspective, emphasizing the pivotal role of manipulatives in facilitating learning within the zone of proximal development. The use of manipulatives aids learners in constructing conceptual understanding by reinforcing abstract ideas. Therefore, the study contributes to existing research highlighting the utility of manipulatives in mathematics classrooms, underscoring their effectiveness in enhancing learning outcomes. [ABSTRACT FROM AUTHOR]

After nationwide school closures due to COVID-19, virtual learning environments (VLE) have seen tremendous increase in usage. The current study identified teacher activities for orchestration using an Algebra VLE during school closures, and whether these activities were related to student achievement. In May 2020, we collected survey data on how 213 teachers were using a VLE for Algebra with 10,590 students, along with system logs and student achievement data. Results indicated that teachers made several changes to teacher strategies due to school closures, including allowing students more time to complete assignments. Multilevel modeling showed that teacher orchestration activities, particularly those related to regulation/management and awareness/assessment, were positively related to student achievement. We discussed the results and provided implications for practice (Q&A setting, assignment flexibility). [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]

Given a Hopf algebra H and a counital 2-cocycle μ on H, Drinfeld introduced a notion of twist which deforms an H-module algebra A into a new algebra A μ . We show that when A is a quadratic algebra, and H acts on A by degree-preserving endomorphisms, then the twist A μ is also quadratic. Furthermore, if A is a Koszul algebra, then A μ is a Koszul algebra. As an application, we prove that the twist of the q-quantum plane by the quasitriangular structure of the quantum enveloping algebra U q (s l 2) is a quadratic algebra equal to the q − 1 -quantum plane. [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]

There is a well-known construction of a Jordan algebra via a sharped cubic form. We introduce a generalized sharped cubic form and prove that the split spin factor algebra is induced by this construction, and it satisfies the identity ((a , b , c) , d , b) + ((c , b , d) , a , b) + ((d , b , a) , c , b) = 0 . The split spin factor algebras have recently appeared in the classification of 2-generated axial algebras of Monster type fulfilled by T. Yabe; their properties were studied by J. McInroy and S. Shpectorov. [ABSTRACT FROM AUTHOR]

We obtain criteria for when a ring with enough idempotents is left/right artinian or noetherian in terms of local criteria defined by the associated complete set of idempotents for the ring. We apply these criteria to object unital category graded rings in general and, in particular, to the class of skew category algebras. Thereby, we generalize results by Nastasescu-van Oystaeyen, Bell, Park, and Zelmanov from the group graded case to groupoid, and in some cases category, gradings. [ABSTRACT FROM AUTHOR]

This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra Kn, n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, we give the description of the finite-dimensional Nichols algebras over simple modules over K3. This includes a family of 12-dimensional Nichols algebras { B ξ } depending on 3rd roots of unity. Here, B 1 is isomorphic to the well-known Fomin-Kirillov algebra, and B ξ ≃ B ξ 2 as graded algebras but B 1 is not isomorphic to B ξ as algebra for ξ ≠ 1 . As a byproduct we obtain new Hopf algebras of dimension 216. [ABSTRACT FROM AUTHOR]

We construct 2-step nilpotent Lie algebras using labeled directed simple graphs and give a criterion to detect certain ideals and subalgebras by finding special subgraphs. We prove that if a label occurs only once, then reversing its orientation leads to an isomorphic algebra. As a consequence, if every edge is labeled differently, the Lie algebra depends only on the underlying undirected graph. In addition, we construct the graphs of all 2-step nilpotent Lie algebras of dimension ≤ 6 and compute the algebra of strata preserving derivations of the Lie algebra associated with the complete bipartite graph K m , n with two different labelings. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a basis for the center of the infinitesimal q-Schur algebra s q (2 , r) 1 in terms of its certain basis. [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]

The number of undergraduates placing into developmental or "remedial" coursework continues to increase. This rise is concerning because developmental or "remedial" coursework creates barriers for many students. This mixed method study examines a holistic approach to placement. After one-on-one interviews with mathematics faculty, students (N = 8) within two points of passing our mathematics placement test (MPT) were given the opportunity to enroll in college-level coursework (Business Calculus) instead of the developmental coursework (Algebra). There were no significant differences in final exam and course grades between these students and those who initially placed into the course. Additionally, the qualitative data revealed many common themes and future ideas for intervention. [ABSTRACT FROM AUTHOR]

In this paper, we examine the relationships between teachers' subject matter preparation and experience in teaching and their performance on an instrument measuring mathematical knowledge for teaching Algebra 1. We administered the same instrument to two different samples of teachers−high school practicing teachers and community college faculty − who teach the same algebra content in different levels of institutions, and we compared the performance of the two different samples and the relationships between the measured knowledge and their educational and teaching background across the samples. The comparison suggested that the community college faculty possess a higher level of mathematical knowledge for teaching Algebra 1 than high school teachers. The subsequent analyses using the Multiple Indicator Multiple Causes (MIMIC) models based on our hypothesis on the factors contributing to the differences in the knowledge between the two teacher samples suggest that experience teaching advanced algebra courses has positive effects on the mathematical knowledge for teaching Algebra 1 in both groups. Highlighting the positive effect of algebra-based teaching experience on test performance, we discuss the implications of the impact of subject specific experience in teaching on teachers' mathematical content knowledge for teaching. [ABSTRACT FROM AUTHOR]

Let be a nontrivial nest on a separable Hilbert space and be the associated nest algebra. In this note, we prove that, if contains an atom of infinite rank or is uncountable, then the unit operator I is a Lie all-derivable point of. [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]

Let FI(X, K) be the finitary incidence algebra of a poset X over a field K. In this short note we establish when FI(X, K) satisfies a polynomial identity and when its group of units U (F I (X , K)) satisfies a group identity. The Lie derived length of FI(X, K) and the derived length of U (F I (X , K)) are also determined. [ABSTRACT FROM AUTHOR]

Let R be a left-symmetric conformal algebra and Q be a C [ ∂ ] -module. We introduce the notion of a unified product for left-symmetric conformal algebras and apply it to construct an object H R 2 (Q , R) to describe and classify all left-symmetric conformal algebra structures on the direct sum E = R ⊕ Q as a C [ ∂ ] -module such that R is a subalgebra of E up to isomorphism whose restriction on R is the identity map. Moreover, we study H R 2 (Q , R) in detail when Q, R are free as C [ ∂ ] -modules and rank Q = 1 . Some special products such as crossed product and bicrossed product are also investigated. [ABSTRACT FROM AUTHOR]

The Frobenius-Perron dimension of a matrix, also known as the spectral radius, is a useful tool for studying linear algebras and plays an important role in the classification of the representation categories of algebras. In this paper, we study the Frobenius-Perron theory of the representation categories of bound quiver algebras containing loops, and find a way to calculate the Frobenius-Perron dimensions of these algebras satisfying the commutativity condition of loops. As an application, we prove that the Frobenius-Perron dimension of the representation category of a modified ADE bounded quiver algebra is equal to the maximal number of loops at each vertex. Finally, we point out that there also exist infinite dimensional algebras whose Frobenius-Perron dimensions is equal to the maximal number of loops by giving an example. [ABSTRACT FROM AUTHOR]

It is classical that certain truncations of the translation quiver Z Q appear in the Auslander-Reiten quiver. The stable n-translation quiver Z | n − 1 Q is introduced as a generalization of Z Q in studying higher representation theory. In this paper, we find conditions for a Hom-finite k-category to be realized as the bound path category of a convex full subquiver of a stable n-translation quiver. We show that the n-preinjective component and n-preprojective component of n-slice algebra are realized as the bound path categories of some truncations of Z | n − 1 Q o p . We also use Z | n − 1 Q o p to describe the νn-closure of Γ in the derived category. [ABSTRACT FROM AUTHOR]

Let k be a field and q ∈ k * not a root of unity. We prove that on the quantum plane algebra P q (k) = k q [ x , y ] any local derivation is a derivation. [ABSTRACT FROM AUTHOR]

The goal of this paper is to construct a semifree resolution for a non-negatively graded strongly commutative DG algebra B over the enveloping DG algebra B ⊗ A B , where A ⊆ B is a DG subalgebra and B is semifree over A. Our construction of such a semifree resolution, that we denote by (B , D) , uses the notions of reduced bar resolution and tensor algebra of the shift of the diagonal ideal. As an application of (B , D) , we provide a new characterization of naïve liftability of semifree DG B-modules. [ABSTRACT FROM AUTHOR]

A classical theorem of Wonenburger, Djokovič, Hoffmann and Paige [2, 3, 7] states that an element of the general linear group of a finite-dimensional vector space is the product of two involutions if and only if it is similar to its inverse. We give a very elementary proof of this result when the underlying field F is algebraically closed with characteristic other than 2. In that situation, the result is generalized to the group of invertibles of any finite-dimensional algebra over F. [ABSTRACT FROM AUTHOR]

In the study of pre-Lie algebras, the concept of pre-morphism arises naturally as a generalization of the standard notion of morphism. Pre-morphisms can be defined for arbitrary (not-necessarily associative) algebras over any commutative ring k with identity, and can be dualized in various ways to generalized morphisms (related to pre-Jordan algebras) and anti-pre-morphisms (related to anti-pre-Lie algebras). We consider idempotent pre-endomorphisms (generalized endomorphisms, anti-pre-endomorphisms). Idempotent pre-endomorphisms are related to semidirect-product decompositions of the sub-adjacent anticommutative algebra. [ABSTRACT FROM AUTHOR]

Let M be the free metabelian Leibniz algebra of rank two over a field K of characteristic 0. In this article, we show that an endomorphism of M which preserves the orbit of a nontrivial certain kind of element w ∈ M ′ is an automorphism. Additionally, we give some test elements. [ABSTRACT FROM AUTHOR]

In this paper, we define a new cohomology theory for multiplicative Hom-pre-Lie algebras which controls deformations of Hom-pre-Lie algebra structure. This new cohomology is a natural one by considering the structure map. We develop equivariant cohomology theory for a Hom-pre-Lie algebra equipped with a finite group action by formulating a proper notion of coefficients system for the equivariant cohomology. We also study the associated formal deformation theory for Hom-pre-Lie algebras in the equivariant context. [ABSTRACT FROM AUTHOR]

The recursion formula of the quotient difference algorithm for computing matrix eigenvalues corresponds with the discrete Toda equation, which is well-known in discrete integrable systems. Previous studies have revealed that the ultradiscrete Toda equation computes eigenvalues of tridiagonal matrices over the min-plus algebra. The min-plus algebra is a commutative semiring in which minimum and plus operations are introduced into the union of the set of real numbers and positive infinity. The discrete hungry Toda equation, which is a generalization of the discrete Toda equation, can compute the eigenvalues of lower Hessenberg banded matrices. This study focuses on the ultradiscrete hungry Toda equation and show that the time evolution of the equation yields the eigenvalues of lower Hessenberg banded matrices over the min-plus 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]

We study realizable values of the length function for unital possibly nonassociative algebras of a given dimension. To do this we apply the method of characteristic sequences and establish sufficient conditions of realizability for a given value of length. The proposed conditions are based on binary decompositions of the value and algebraic constructions that allow to modify length function of an algebra. Additionally we provide a description of unital algebras of maximal possible length in terms of their bases. [ABSTRACT FROM AUTHOR]