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In this paper, we obtain, respectively, some new linear bases of free nonunitary (modified) weighted differential algebras and free nonunitary (modified) Rota–Baxter algebras, in terms of the method of Gröbner–Shirshov bases. [ABSTRACT FROM AUTHOR]

In this paper, we shed new light on the Flexible Atom Conjecture. We first give finite representation results for relation algebras 3 3 3 7 , 3 5 3 7 , 7 7 8 3 , 7 8 8 3 , 8 0 8 3 , 8 2 8 3 , 8 3 8 3 , 1 3 1 0 1 3 1 6 , 1 3 1 3 1 3 1 6 , 1 3 1 5 1 3 1 6 and 1 3 1 6 1 3 1 6 . Prior to our paper, only 8 3 8 3 and 1 3 1 6 1 3 1 6 were known to be finitely representable. We accomplish this by generalizing the notion of a relation algebra generated by a Ramsey scheme to the directed (antisymmetric) setting, and then showing that each of these algebras embeds into a finite directed anti-Ramsey scheme. The notion of a directed anti-Ramsey scheme may be of independent interest. We complement our upper bounds with some lower bounds. Namely, we show that any square representation of 3 1 3 7 requires at least 14 points, any square representation of 3 3 3 7 requires at least 11 points, and any square representation of 3 5 3 7 requires at least 12 points. Our technique adapts previous work of Alm et al. [Algebra Univ. (2022)], in that we examine the combinatorial structure induced by the flexible atom. [ABSTRACT FROM AUTHOR]

In this paper, we present a method for finding the absorbing radical of a finite-dimensional evolution algebra. Such a method consists of finding the acyclic vertices of an oriented graph associated with the algebra. The set of generators associated with such vertices turn out to be the generators of the absorption radical. As an application we use the absorption radical to study the decomposability of some degenerate evolution algebras. [ABSTRACT FROM AUTHOR]

In this paper, we prove that every matrix over a division ring is representable as a product of at most 10 traceless matrices as well as a product of at most four semi-traceless matrices. By applying this result and the obtained so far other results, we show that elements of some algebras possess some rather interesting and nontrivial decompositions into products of images of non-commutative polynomials. [ABSTRACT FROM AUTHOR]

Let (q) be a finite-dimensional nilpotent algebra over a finite field q with q elements, and let G (q) = 1 + (q). On the other hand, let denote the algebraic closure of q , and let = (q) ⊗ q . Then G = 1 + is an algebraic group over equipped with an q -rational structure given by the usual Frobenius map F : G → G , and G (q) can be regarded as the fixed point subgroup G F . For every n ∈ ℕ , the nth power F n : G → G is also a Frobenius map, and G F n identifies with G (q n) = 1 + (q n). The Frobenius map restricts to a group automorphism F : G (q n) → G (q n) , and hence it acts on the set of irreducible characters of G (q n). Shintani descent provides a method to compare F-invariant irreducible characters of G (q n) and irreducible characters of G (q). In this paper, we show that it also provides a uniform way of studying supercharacters of G (q n) for n ∈ ℕ. These groups form an inductive system with respect to the inclusion maps G (q m) → G (q n) whenever m | n , and this fact allows us to study all supercharacter theories simultaneously, to establish connections between them, and to relate them to the algebraic group G. Indeed, we show that Shintani descent permits the definition of a certain "superdual algebra" which encodes information about the supercharacters of G (q n) for n ∈ ℕ. [ABSTRACT FROM AUTHOR]

Let F be an algebraically closed field of characteristic zero and let A be a minimal superalgebra with semisimple part A s s = A 1 ⊕ ⋯ ⊕ A n , where each A i is a simple superalgebra. We say that A is r -special if n ≥ 3 and there exists 1 < r < n such that A r = M k r (F ⊕ c F) , with c 2 = 1 , whereas, for the remaining indices s , A s = M k s , l s (F). In [1], Avelar et al. classified the r -special minimal superalgebras and described some necessary and sufficient conditions for the varieties generated by such superalgebras to be minimal of ℤ 2 -exponent dim F (A 1 ⊕ ⋯ ⊕ A n). In this paper, we give continuity to a such study and we made explicit that the ℤ 2 -gradings play an important role in the minimality of some varieties. We provide sufficient conditions, depending on the ℤ 2 -grading of A , for the variety generated by A to be minimal. Moreover, we characterize two families of (n − 1) -special minimal superalgebras and we give necessary and sufficient conditions (depending on their ℤ 2 -gradings) for the varieties generated by them to be minimal. [ABSTRACT FROM AUTHOR]

Kudryavtseva and Mazorchuk exhibited Schur–Weyl duality between the rook monoid algebra ℂ R n and the subalgebra ℂ I k of the partition algebra ℂ A k (n) acting on (ℂ n) ⊗ k . In this paper, we consider a subalgebra ℂ I k + 1 2 of ℂ I k + 1 such that there is Schur–Weyl duality between the actions of ℂ R n − 1 and ℂ I k + 1 2 on (ℂ n) ⊗ k . This paper studies the representation theory of partition algebras ℂ I k and ℂ I k + 1 2 for rook monoids inductively by considering the multiplicity free tower ℂ I 1 ⊂ ℂ I 3 2 ⊂ ℂ I 2 ⊂ ⋯ ⊂ ℂ I k ⊂ ℂ I k + 1 2 ⊂ ⋯. Furthermore, this inductive approach is established as a spectral approach by describing the Jucys–Murphy elements and their actions on the canonical Gelfand–Tsetlin bases, determined by the aforementioned multiplicity free tower, of irreducible representations of ℂ I k and ℂ I k + 1 2 . Also, we describe the Jucys–Murphy elements of ℂ R n which play a central role in the demonstration of the actions of Jucys–Murphy elements of ℂ I k and ℂ I k + 1 2 . [ABSTRACT FROM AUTHOR]

Several structural properties of an algebraic structure can be seen from the higher commutators of its congruences. Even on a finite algebra, the sequence of higher commutator operations is an infinite object. In this paper, we exhibit finite representations of this sequence for finite algebras from congruence modular varieties. [ABSTRACT FROM AUTHOR]

A subdirect product A ≤ Π 〈 A i : i ∈ I 〉 is global if there is a topology on I which makes of A the algebra of all global sections of a sheaf whose stalks are the algebras A i with i ∈ I. A quasivariety has BL-global representations if there is a class ℱ containing but close to the class of all relatively subdirectly irreducible members of , satisfying that every member of is isomorphic to a global subdirect product whose factors are in ℱ. The adjective "BL" refers to "Birkhoff-like" since this type of representations are analogous to the classical Birkhoff subdirect representation by relatively subdirectly irreducibles. This paper has two main contributions. The first one is a generalization of a theorem proved in [D. Vaggione, Sheaf representation and Chinese remainder theorems, Algebra Universalis 29 (1992) 232–272] which characterizes the existence of a global representation of an algebra in terms of the solvability of certain congruence systems. The second one is a theorem assuring the existence of BL-global representations for quasivarieties with a near unanimity term. [ABSTRACT FROM AUTHOR]

Let E be the infinite-dimensional Grassmann algebra over an infinite field F of characteristic different from 2. The main purpose of this paper is to describe the T ℤ -ideal of the graded polynomial identities and the T ℤ -space of the central polynomials of E equipped with its 2 and 3-induced ℤ -gradings. Therefore, we generalize the results of [A. Brandão, C. Fidelis and A. Guimarães, ℤ -gradings of full support on the Grassmann algebra, J. Algebra 601 (2022) 332–353; C. Fidelis, A. Guimarães and P. Koshlukov, A note on ℤ -gradings on the Grassmann algebra and elementary number theory, Linear Multilinear Algebra 71(7) (2023) 1244–1264] in the PI theory context. [ABSTRACT FROM AUTHOR]

In this paper, we explore some applications of a certain technique (that we call Freese' technique), which is a tool for identifying certain lattices as sublattices of the congruence lattice of a given algebra. In particular we will give sufficient conditions for two family of lattices (called the rods and the snakes) to be admissible as sublattices of a variety generated by a given algebra, extending an unpublished result of Freese and Lipparini. [ABSTRACT FROM AUTHOR]

In this paper, we show that conjugacy classes of classical Weyl groups W (B n) and W (D n) are of type D. Consequently, we obtain that Nichols algebras of irreducible Yetter–Drinfeld modules over the classical Weyl groups n (n ≥ 5) are infinite dimensional. [ABSTRACT FROM AUTHOR]

For some Maltsev conditions Σ it is enough to check if a finite algebra A satisfies Σ locally on subsets of bounded size in order to decide whether A satisfies Σ (globally). This local–global property is the main known source of tractability results for deciding Maltsev conditions. In this paper, we investigate the local–global property for the existence of a G-term, i.e. an n -ary term that is invariant under permuting its variables according to a permutation group G ≤ Sym (n). Our results imply in particular that all cyclic loop conditions (in the sense of Bodirsky, Starke, and Vucaj) have the local–global property (and thus can be decided in polynomial time), while symmetric terms of arity n > 2 fail to have it. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce and investigate the notion of derivation for quandle algebras. More precisely, we describe the symmetries on structure constants providing a characterization for a linear map to be a derivation. We obtain a complete characterization of derivations in the case of quandle algebras of dihedral quandles over fields of characteristic zero, and provide the dimensionality of the Lie algebra of derivations. Many explicit examples and computations are given over both zero and positive characteristic. Furthermore, we investigate inner derivations, in the sense of Schafer for non-associative structures. We obtain necessary conditions for the Lie transformation algebra of quandle algebras of Alexander quandles, with explicit computations in low dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we calculate the low order relations of B ∞ -algebra and introduce the bibrace algebra. It can be applied to the B ∞ -algebras of the (co)Hochschild cochain complex and the singular Hochschild complex of an algebra. [ABSTRACT FROM AUTHOR]

In this paper, we study the differential smoothness of 3-dimensional skew polynomial algebras and diffusion algebras. [ABSTRACT FROM AUTHOR]

This paper is devoted to the so-called complete Leibniz algebras. We construct some complete Leibniz algebras with complete radical and prove that the direct sum of complete Leibniz algebras is also complete. It is known that a Lie algebra with a complete ideal is split. We discuss the analogs of this result for the Leibniz algebras and show that it is true for some special classes of Leibniz algebras. Finally, we consider derivations of Leibniz algebras and present some classes of Leibniz algebras which are not complete, since they admit outer derivation. [ABSTRACT FROM AUTHOR]

We continue developing the theory of nilpotent symplectic alternating algebras. The algebras of upper bound nilpotent class, that we call maximal algebras, have been introduced and well studied. In this paper, we continue with the external case problem of algebras of minimal nilpotent class. We show the existence of a subclass of algebras over a field F that is of certain lower bound class that depends on the dimension only. This suggests a minimal bound for the class of nilpotent algebras of dimension 2 n ≥ 8 of rank 2 over any field. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic zero and M n (F) the algebra of n × n matrices over F. By the classical Amitsur–Levitzki theorem, it is well known that 2 n is the smallest degree of a standard polynomial identity of M n (F). A theorem due to Rowen shows that when the symplectic involution s is considered, the standard polynomial of degree 4 m − 2 in symmetric variables is an identity of (M 2 m (F) , s). This means that when only certain kinds of matrices are considered in the substitutions, the minimal degree of a standard identity may not remain being the same. In this paper, we present some results about the minimal degree of standard identities in skew or symmetric variables of odd degree of M n (F) in the symplectic graded involution case. Along the way, we also present the minimal total degree of a double Capelli polynomial identity in symmetric variables of M n (F) with transpose involution. [ABSTRACT FROM AUTHOR]

In [A.-P. Grecianu, A. Kvaschuk, A. G. Myasnikov and D. Serbin, Groups acting on hyperbolic Λ -metric spaces, Int. J. Algebra Comput. 25(6) (2015) 977–1042], the authors initiated a systematic study of hyperbolic Λ -metric spaces, where Λ is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case Λ = ℤ n taken with the right lexicographic order and studies the structure of finitely generated groups acting on hyperbolic ℤ n -metric spaces. Under certain constraints, the structure of such groups is described in terms of a hierarchy (see [D. T. Wise, The Structure of Groups with a Quasiconvex Hierarchy : (AMS- 2 0 9) , Annals of Mathematics Studies (Princeton University Press, 2021)]) similar to the one established for ℤ n -free groups in [O. Kharlampovich, A. G. Myasnikov, V. N. Remeslennikov and D. Serbin, Groups with free regular length functions in ℤ n , Trans. Amer. Math. Soc. 364 (2012) 2847–2882]. [ABSTRACT FROM AUTHOR]

Let X and Y be two generic traceless matrices of size 2 × 2 with entries from a commutative associative polynomial algebra over a field K of characteristic zero. Consider the associative unitary algebra W , and its Lie subalgebra L generated by X and Y over the field K. It is well known that the center C (W) = K [ t , u , v ] of W is the polynomial algebra generated by the algebraically independent commuting elements t = tr (X 2) I 2 , u = tr (Y 2) I 2 , v = tr (X Y) I 2 . We call a polynomial p ∈ W symmetric, if p (X , Y) = p (Y , X). The set of symmetric polynomials is equal to the algebra W S 2 of invariants of symmetric group S 2 . Similarly, we define the Lie algebra L S 2 of symmetric polynomials in the Lie algebra L. We give the description of the algebras W S 2 and L S 2 , and we provide finite sets of free generators for W S 2 , and [ L , L ] S 2 as K [ t + u , t u , v ] -modules. [ABSTRACT FROM AUTHOR]

We contribute to a recent research program which aims at revisiting the study of the complexity of word problems, a major area of research in combinatorial algebra, through the lens of the theory of computably enumerable equivalence relations (ceers), which has considerably grown in recent times. To pursue our analysis, we rely on the most popular way of assessing the complexity of ceers, that is via computable reducibility on equivalence relations, and its corresponding degree structure (the c -degrees). On the negative side, building on previous work of Kasymov and Khoussainov, we individuate a collection of c -degrees of ceers which cannot be realized by the word problem of any finitely generated algebra of finite type. On the positive side, we show that word problems of finitely generated semigroups realize a collection of c -degrees which embeds rich structures and is large in several reasonable ways. [ABSTRACT FROM AUTHOR]

We prove that every derivation and every locally nilpotent derivation of the subalgebra K [ x n , x n − 1 y , ... , x y n − 1 , y n ] , where n ≥ 2 , of the polynomial algebra K [ x , y ] in two variables over a field K of characteristic zero is induced by a derivation and a locally nilpotent derivation of K [ x , y ] , respectively. Moreover, we prove that every automorphism of K [ x n , x n − 1 y , ... , x y n − 1 , y n ] over an algebraically closed field K of characteristic zero is induced by an automorphism of K [ x , y ]. We also show that the group of automorphisms of K [ x n , x n − 1 y , ... , x y n − 1 , y n ] admits an amalgamated free product structure. [ABSTRACT FROM AUTHOR]

For an algebra = (A , f 1 , ... , f n) the algebra = (A , t 1 , ... , t k) is called a functional reduct if each t j is a term function of . We classify the functional reducts of the countable atomless Boolean algebra up to first-order interdefinability. That is, we consider two functional reducts the "same" if their group of automorphisms is the same. We show that there are 13 such reducts and describe their structures and group of automorphisms. [ABSTRACT FROM AUTHOR]

We study rigidity questions for pairs of Lie algebras (,) admitting a post-Lie algebra structure. We show that if is semisimple and is arbitrary, then we have rigidity in the sense that and must be isomorphic. The proof uses a result on the decomposition of a Lie algebra = 1 ∔ 2 as the direct vector space sum of two semisimple subalgebras. We show that must be semisimple and hence isomorphic to the direct Lie algebra sum ≅ 1 ⊕ 2 . This solves some open existence questions for post-Lie algebra structures on pairs of Lie algebras (,). We prove additional existence results for pairs (,) , where is complete, and for pairs, where is reductive with 1 -dimensional center and is solvable or nilpotent. [ABSTRACT FROM AUTHOR]

We study prime and semiprime Novikov algebras. We prove that prime nonassociative Novikov algebra has zero nucleus and center. It is well known that an ideal of an alternative semiprime algebra is semiprime algebra. We proved this statement for Novikov algebras. It implies that a Baer radical exists in a class of Novikov algebras. [ABSTRACT FROM AUTHOR]

We investigate the class of finite-dimensional not necessarily associative algebras that have slowly growing length, that is, for any algebra in this class its length is less than or equal to its dimension. We show that this class is considerably big, in particular, finite-dimensional Lie algebras as well as many other important classical finite-dimensional algebras belong to this class, for example, Leibniz algebras, Novikov algebras and Zinbiel algebras. The exact upper bounds for the length of these algebras is proved. To do this, we transfer the method of characteristic sequences to non-unital algebras and find certain polynomial conditions on the algebra elements that guarantee the slow growth of the length function. [ABSTRACT FROM AUTHOR]

We prove that the class of algebras embeddable into Leibniz algebras with respect to the commutator product is not a variety. It is shown that every commutative metabelain algebra is embeddable into Leibniz algebras with respect to the anti-commutator. Furthermore, we study polynomial identities satisfied by the commutator in every Leibniz algebra. We extend the result of Dzhumadil'daev in [A. S. Dzhumadil'daev, q-Leibniz algebras, Serdica Math. J. 34(2) (2008) 415–440]. to identities up to degree 7 and give a conjecture on identities of higher degrees. As a consequence, we obtain an example of a non-Spechtian variety of anticommutative algebras. [ABSTRACT FROM AUTHOR]

In this note, we consider low-dimensional metric Leibniz algebras with an invariant inner product over the complex numbers up to dimension 5. We study their deformations, and give explicit formulas for the cocycles and deformations. We identify among those the metric deformations. [ABSTRACT FROM AUTHOR]

We study (support) τ -tilting modules over the trivial extensions of finite-dimensional algebras. More precisely, we construct two classes of (support) τ -tilting modules in terms of the adjoint functors which extend and generalize the results on (support) τ -tilting modules over triangular matrix rings given by Gao-Huang. [ABSTRACT FROM AUTHOR]

Let P and Q be finite posets and R a commutative unital ring. In the case where R is indecomposable, we prove that the R -linear isomorphisms between partial flag incidence algebras I 3 (P , R) and I 3 (Q , R) are exactly those induced by poset isomorphisms between P and Q. We also show that the R -linear derivations of I 3 (P , R) are trivial. [ABSTRACT FROM AUTHOR]

Let I (P) denote the incidence algebra of a locally finite poset (P , ≤) over a field F and ∼ some equivalence relation on the set of generators of I (P). Then I (P , ∼) is the subset of I (P) of all the elements that are constant on the equivalence classes of ∼. If ∼ satisfies certain conditions, then I (P , ∼) is a subalgebra of I (P) called a reduced incidence algebra. We extend this notion to finitary incidence algebras F I (P) for any poset (P , ≤). We investigate reduced finitary incidence algebras F I (P , ∼) and determine their automorphisms in some special cases. [ABSTRACT FROM AUTHOR]

For each integer n ≥ 0 , we define a category whose objects are discriminant algebra functors in rank n , namely, choices of how to attach functorially to each rank- n algebra a quadratic algebra with the same discriminant. We show that the discriminant algebra functors defined by Loos, Rost, and the present authors are all isomorphic in this category, and prove furthermore that in ranks n ≤ 3 discriminant algebra functors are unique up to unique isomorphism. [ABSTRACT FROM AUTHOR]

If E is a directed graph and K is a field, the Leavitt path algebra L K (E) of E over K is naturally graded by the group of integers ℤ. We formulate properties of the graph E which are equivalent with L K (E) being a crossed product, a skew group ring, or a group ring with respect to this natural grading. We state this main result so that the algebra properties of L K (E) are also characterized in terms of the pre-ordered group properties of the Grothendieck ℤ -group of L K (E). If E has finitely many vertices, we characterize when L K (E) is strongly graded in terms of the properties of K 0 Γ (L K (E)). Our proof also provides an alternative to the known proof of the equivalence L K (E) is strongly graded if and only if E has no sinks for a finite graph E. We also show that, if unital, the algebra L K (E) is strongly graded and graded unit-regular if and only if L K (E) is a crossed product. In the process of showing the main result, we obtain conditions on a group Γ and a Γ -graded division ring K equivalent with the requirements that a Γ -graded matrix ring R over K is strongly graded, a crossed product, a skew group ring, or a group ring. We characterize these properties also in terms of the action of the group Γ on the Grothendieck Γ -group K 0 Γ (R). [ABSTRACT FROM AUTHOR]

The aim of this work is to provide a concrete list of non-isomorphic commutative power-associative superalgebras up to dimension 4 over an algebraically closed field of characteristic prime to 30. As a byproduct, we exhibit an example of a simple non-Jordan power-associative superalgebra whose even part is not semisimple. [ABSTRACT FROM AUTHOR]

We show that the right ideal of a Novikov algebra generated by the square of a right nilpotent subalgebra is nilpotent. We also prove that a G -graded Novikov algebra N over a field K with solvable 0 -component N 0 is solvable, where G is a finite additive abelean group and the characteristic of K does not divide the order of the group G. We also show that any Novikov algebra N with a finite solvable group of automorphisms G is solvable if the algebra of invariants N G is solvable. [ABSTRACT FROM AUTHOR]