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A set of position-dependent noncommutative algebra in two dimension (2D) that describes the space near the Planck scale had been introduced [J. Phys. A: Math. Theor. 53 (2020) 115303]. This algebra predicted the existence of maximal length of graviton measurable at low energy. From this algebra, we deduce in the present paper, a new noncommutative algebra that is compatible with the deformed algebra proposed by Costa Filho et al. [Phys. Rev. A84 (2011) 050102] to describe the Position-Dependent Mass (PDM) system in 1D. To this aim, we derive from the momentum operators, the Schrödinger-like equation which describes PDM system in null quantum well potential. The spectrum of this system is asymetrically deformed and exhibits different behaviours from the one obtained by Costa Filho et al. Thus, we observe that by increasing the PDM, the energy decreases and this decrease is more pronounced as the quantum number increases. Finally, we evaluate the thermodynamic quantities within the canonical ensemble and we show that these results are consistent with the ones recently obtained by Bensalem and Bouaziz [Phys. A Stat. Mech. Appl.523 (2019) 583–592]. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic zero and x a free variable. A K - ℰ -derivation of K [ x ] is a K -linear map of the form I , − , ϕ for some K -algebra endomorphism ϕ of K [ x ] , where I denotes the identity map of K [ x ]. In this paper, we study the image of an ideal of K [ x ] under some K -derivations and K - ℰ -derivations of K [ x ]. We show that the LFED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and ℰ -derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all K - ℰ -derivations and all locally finite K -derivations of K [ x ]. We also show that the LNED conjecture proposed in [W. Zhao, Some open problems on locally finite or locally nilpotent derivations and ℰ -derivations, Commun. Contem. Math. 20(4) (2018) 1750056] holds for all locally nilpotent K -derivations of K [ x ] , and also for all locally nilpotent K - ℰ -derivations of K [ x ] and the ideals u K [ x ] such that either u = 0 , or deg u ≤ 1 , or u has at least one repeated root in the algebraic closure of K. As a bi-product, the homogeneous Mathieu subspaces (Mathieu–Zhao spaces) of the univariate polynomial algebra over an arbitrary field have also been classified. [ABSTRACT FROM AUTHOR]

For an ample groupoid and a unit x of , Steinberg constructed the induction and restriction functors between the category of modules over the Steinberg algebra A R () and the category of modules over the isotropy group algebra R x . In this paper, we prove a graded version of these functors and give a description of spectral graded simple modules over the graded Steinberg algebra A R (). As an application, the spectral simple and graded simple modules over the Leavitt path algebra L K (E) are classified. In particular, we show that many of previously known simple and graded simple L K (E) -modules, including the Chen simple modules, are induced from (ungraded or graded) simple modules over isotropy group algebras. [ABSTRACT FROM AUTHOR]

Given a Hopf algebra H , Brzeziński and Militaru have shown that each braided commutative Yetter–Drinfeld H -module algebra A gives rise to an associative A -bialgebroid structure on the smash product algebra A ♯ H. They also exhibited an antipode map making A ♯ H the total algebra of a Lu's Hopf algebroid over A. However, the published proof that the antipode is an antihomomorphism covers only a special case. In this paper, a complete proof of the antihomomorphism property is exhibited. Moreover, a new generalized version of the construction is provided. Its input is a compatible pair A and A op of braided commutative Yetter–Drinfeld H -module algebras, and output is a symmetric Hopf algebroid A ♯ H ≅ H ♯ A op over  A. This construction does not require that the antipode of H is invertible. [ABSTRACT FROM AUTHOR]

In this paper, we construct the super Virasoro algebra with an arbitrary conformal dimension Δ from the generalized ℛ (p , q) -deformed quantum algebra and investigate the ℛ (p , q) -deformed super Virasoro algebra with the particular conformal dimension Δ = 1. Furthermore, we perform the ℛ (p , q) -conformal Virasoro n -algebra, the ℛ (p , q) -conformal super Virasoro n -algebra (n -even) and discuss a toy model for the ℛ (p , q) -conformal Virasoro constraints and ℛ (p , q) -conformal super Virasoro constraints. Besides, we generalized the notion of the ℛ (p , q) -elliptic Hermitian matrix model with an arbitrary conformal dimension Δ. Finally, we deduce relevant particular cases generated by quantum algebras known in the literature. [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]

In this paper, we introduce the notion of essential extensions of fuzzy modules. We use these concepts to introduce the notion of injective hulls of fuzzy modules. It is known that every R-module has an injective hull, where R is a ring. We show that these corresponding results do not hold for fuzzy R-modules, i.e. there exists a fuzzy R-module that does not have an injective hull. Sufficient conditions are given for a fuzzy R-module to have an injective hull. [ABSTRACT FROM AUTHOR]

Action of a pomonoid on partially ordered sets (S -posets) has beautiful aspects in practical subjects such as automata theory, projection algebra and theoretical computer science which makes it always capture the interest of mathematicians. On the other hand, the study of different kinds of weakly injectivity (which category theory inherited from homological and commutative algebra) is an interesting subject for mathematicians. One of the important kinds of weakly injectivity is quasi-injectivity. In this paper, we study quasi-injectivity in the category of S -posets with respect to special kind of order embeddings, namely, down-closed embeddings (dc-quasi-injectivity). [ABSTRACT FROM AUTHOR]

In this paper, we continue the study of Hurwitz series over a commutative unital ring that was begun in [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304]. In particular, we introduce the notion of multivariate Hurwitz series. The underlying idea is that multivariate Hurwitz series are to Hurwitz series as studied in [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304] as formal power series in several indeterminates are to formal power series in only one indeterminate. The elementary aspects of the theory follow along the lines of [W. Keigher, On the ring of Hurwitz series, Commun. Algebra 25 (1997) 1845–1859; W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304]. The treatment of substitution and divided powers introduces special problems not encountered in [W. Keigher and F. Pritchard, Hurwitz series as formal functions, J. Pure Appl. Algebra 146 (2000) 291–304] and requires special attention to subtle details. However, we are able to establish analogous results. With substitution and divided powers in place, we construct and study the analog to the so called inner transformations of [S. Bochner and W. T. Martin, Several Complex Variables (Princeton University Press, 1948)]. Finally, we are able to establish analogs to many of the fundamental results of single and multivariate calculus. [ABSTRACT FROM AUTHOR]

Operated algebras, that is, algebras equipped with linear operators, have important applications in mathematics and physics. Two primary instances of operated algebras are the Rota–Baxter algebra and TD-algebra. In this paper, we introduce a λ -TD algebra that includes both the Rota–Baxter algebra and the TD-algebra. The explicit construction of free commutative λ -TD algebra on a commutative algebra is obtained by a generalized shuffle product, called the λ -TD shuffle product. We then show that the free commutative λ -TD algebra possesses a left counital bialgebra structure by means of a suitable 1-cocycle condition. Furthermore, the classical result that every connected filtered bialgebra is a Hopf algebra, is extended to the context of left counital bialgebras. Given this result, we finally prove that the left counital bialgebra on the free commutative λ -TD algebra is connected and filtered, and thus is a left counital Hopf algebra. [ABSTRACT FROM AUTHOR]

This paper is devoted to defining and studying Whittaker modules and high order Whittaker modules for the N = 1 super-BMS3 algebra. We also classify the simple Whittaker modules and obtain the necessary and sufficient conditions for the irreducibility of these modules. [ABSTRACT FROM AUTHOR]

In this paper, we construct a holomorphic Siegel modular form of weight 2 and level 2, and compute its Fourier coefficients explicitly. Moreover, we prove that this modular form equals the generating function of the representative number ρ (n , m , r) associated with the maximal order in the quaternion algebra (− 1 , − 1) ℚ . As a corollary, we can give a new proof of the famous formula for the sums of three squares. As applications, we give an explicit formula for the numbers of solutions of two systems of Diophantine equations related with Sun's "1-3-5 conjecture". Furthermore, we show that "a perfect square" in the integral condition version of Sun's conjecture can be replaced by "a power of 4". [ABSTRACT FROM AUTHOR]

Two mathematical fields that concentrate on creating and analyzing structures are algebra and graph theory. There are numerous studies linking algebraic structures like groups, rings, fields and vector spaces with graph theory. Several algebraic graphs have been defined based on the properties of the order of the group and its elements. In this paper, we systematically review the literature on such graphs to understand the research dynamics in the field. [ABSTRACT FROM AUTHOR]

This paper explores several topics related to Woodin's HOD conjecture. We improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the first strongly compact cardinal carries an ordinal definable ω -Jónsson algebra. We show that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then every supercompact cardinal is supercompact in HOD. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the notion of Rota–Baxter mock-Lie bialgebras (A , ∘ , δ , K , Q) and their admissibility conditions in terms of dual representations (ρ ∗ , β ∗ , V ∗). Next, we show that Rota–Baxter mock-Lie bialgebras are characterized by matched pairs and Manin triples of Rota–Baxter mock-Lie algebras. Furthermore, the coboundary case leads to the introduction of the admissible mock-Lie Yang–Baxter equation in Rota–Baxter mock-Lie algebras, whose skew-symmetric solutions are used to construct Rota–Baxter mock-Lie bialgebras. [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]

Random hypergraphs and random simplicial complexes on finite vertices were studied by [M. Farber, L. Mead and T. Nowik, Random simplicial complexes, duality and the critical dimension, J. Topol. Anal.41(1) (2022) 1–32]. The map algebra on random sub-hypergraphs of a fixed simplicial complex, which detects relations between random sub-hypergraphs and random simplicial sub-complexes, was studied by the authors of this paper. In this paper, we study the map algebra on random sub-hypergraphs of a fixed hypergraph. We give some algorithms generating random hypergraphs and random simplicial complexes by considering the actions of the map algebra on the space of probability distributions. We prove some Künneth-type formulae for random hypergraphs and random simplicial complexes on finite vertices. [ABSTRACT FROM AUTHOR]

In an unpublished work of Fasel–Rao–Swan the notion of the relative Witt group W E (R , I) is defined. In this paper, we will give the details of this construction. Then we study the injectivity of the relative Vaserstein symbol V R , I : U m 3 (R , I) / E 3 (R , I) → W E (R , I). We establish injectivity of this symbol if R is a non-singular affine algebra of dimension 3 over a perfect C 1 -field and I is a local complete intersection ideal of R. It is natural to ask whether the Vaserstein symbol is injective for a singular 3-dimensional affine algebra. At the end of the paper, we will give an example of a singular 3-dimensional algebra over a perfect C 1 -field for which the Vaserstein symbol is injective. [ABSTRACT FROM AUTHOR]

Monounary algebras are the most simple type of an algebraic structure. Oriented graphs with one outgoing arrow from every vertex represent them. The aim of this paper is to point out the interdisciplinary relationships concerning this structure. Bernoulli shift is a paradigmatic mapping in dynamical systems. It is also called dyadic, bit shift, doubling or sawtooth. We offer a look at the properties of this mapping via monounary algebras. [ABSTRACT FROM AUTHOR]

In this paper, we study the one-parameter κ -generalized oscillator algebra κ (which includes the case of the standard Weyl–Heisenberg algebra). This algebra admits representations of finite-dimensional for negative values of the parameter κ. We construct the associated Perelomov like coherent states. We discuss their statistical properties. In particular, we derive the expressions of the Mandel parameter and Husimi function and we discuss the squeezing of the κ coherent states. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if R is an affine algebra of dimension d ≥ 4 over ¯ p and 1 / (d − 1) ! ∈ R , then any unimodular row over R of length d can be mapped to a factorial row by elementary transformations. [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]

In this paper, we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces L (p , q) , KBSM(L (p , q)), for q ≠ 0. For doing this, we introduce a new concept, that of an unoriented braid. Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the generalized Temperley–Lieb algebra of type B, TL 1 , n , which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket-type invariant, V , for knots and links in ST, via a unique Markov trace constructed on TL 1 , n . The universal invariant V is equivalent to the KBSM(ST). For passing now to the KBSM(L (p , q)), we impose on V relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in L (p , q) but not in ST, and which reflect the surgery description of L (p , q) , obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM(L (p , q)). We first present the solution for the case q = 1 , which corresponds to obtaining a new basis, ℬ p , for KBSM(L (p , 1)) with (⌊ p / 2 ⌋ + 1) elements. We note that the basis ℬ p is different from the one obtained by Hoste and Przytycki. For dealing with the complexity of the infinite system for the case q > 1 , we first show how the new basis ℬ p of KBSM(L (p , 1)) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case q > 1. The advantage of the braid theoretic approach that we propose for computing skein modules of c.c.o. 3-manifolds, is that the use of braids provides more control on the isotopies of knots and links in the manifolds, and much of the diagrammatic complexity is absorbed into the proofs of the algebraic statements. [ABSTRACT FROM AUTHOR]

In this paper, we continue the project of systematic construction of invariant differential operators (IDOs) on the example of the noncompact algebras s p (n , 1). Our choice of these algebras is motivated by the fact that they belong to a narrow class of algebras, which are of split rank one, of which class the other cases were studied, some long time ago. We concentrate on the case n = 2. We give the main multiplets and the main reduced multiplets of indecomposable elementary representations (ER) for, including the necessary data for all relevant invariant differential operators. We also present explicit expressions for the singular vectors and the intertwining differential operators. [ABSTRACT FROM AUTHOR]

This is a continuation of a previous study initiated by the third author on nonlocal vertex bialgebras and smash product nonlocal vertex algebras. In this paper, we study a notion of right H -comodule nonlocal vertex algebra for a nonlocal vertex bialgebra H and give a construction of deformations of vertex algebras with a right H -comodule nonlocal vertex algebra structure and a compatible H -module nonlocal vertex algebra structure. We also give a construction of ϕ -coordinated quasi modules for smash product nonlocal vertex algebras. As an example, we give a family of quantum vertex algebras by deforming the vertex algebras associated to non-degenerate even lattices. [ABSTRACT FROM AUTHOR]

Two presentations of quantum affine superalgebras were introduced by Yamane in [On defining relations of affine Lie superalgebras and affine quantized universal enveloping superalgebras, Publ. Res. Inst. Math. Sci. 35 (1999) 321–390], which were called Drinfeld–Jimbo realization and Drinfeld realization. Drinfeld realization contains infinite sequences of generators and relations. In this paper, we consider the Drinfeld realization of quantum affine superalgebra q ( ̂ m | n) associated to type m | n and define a simple algebra 0 ( ̂ m | n) generated by only a finite part of these sequences of quantum affine superalgebra q ( ̂ m | n). We show that the algebra 0 ( ̂ m | n) is isomorphic to the quantum affine superalgebra q ( ̂ m | n). Using the above isomorphism, we prove there exists an isomorphism between the two realizations. [ABSTRACT FROM AUTHOR]

Let R be a ring of dimension d ≥ 1 containing ℚ and A = R [ X , Y , Z , W ] (X Y − Z W) . This paper examines the question of efficient generation of ideals in A. [ABSTRACT FROM AUTHOR]

Let and ′ be two ∗-Jordan algebras with identities I and I ′ , respectively, and e a nontrivial ∗-idempotent in . In this paper, we study the characterization of multiplicative ∗-Jordan-type maps. In particular, we provide a characterization in the case of unital prime associative algebra endowed with an involution. [ABSTRACT FROM AUTHOR]

A classical theorem of Ramanujan relates an integral of Dedekind eta-function to a special value of a Dirichlet L-function at s = 2. Ahlgren, Berndt, Yee and Zaharescu have generalized this result [1]. In this paper, we generalize this result to the context of holomorphic cusp forms on the upper half space. [ABSTRACT FROM AUTHOR]

This paper details the diassociative analogue of results concerning the Schur multiplier and other extension-theoretic concepts that originate in group theory. We first prove that covers of diassociative algebras are unique. Second, we show that the multiplier of a diassociative algebra is characterized by the second cohomology group with coefficients in the field. Third, we establish criteria for when the center of a cover maps onto the center of the algebra. Along the way, we obtain a collection of exact sequences, characterizations, and a brief theory of unicentral diassociative algebras and stem extensions. This paper is part of an ongoing project to advance extension theory in the context of several Loday algebras. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of 2-nil ideal introduced by Yetkin Celikel in [E. Yetkin Celikel, 2-nil ideals of commutative rings, Bulletin of the Belgian Mathematical Society Simon Stevin 28 (3) (2021).] to 2-nil submodule which is a subclass of 2-absorbing primary submodules. Let M be an R -module and N be a proper submodule of M. We say that N is a 2-nil submodule of M if whenever a , b ∈ R , m ∈ M and a b m ∈ N , then a b ∈ Nil (M) or a m ∈ N or b m ∈ N. We study the properties of this concept and establish several characterizations. We also investigate the 2-nil ideals of amalgamation. The obtained results yield new original families of examples of 2-nil ideals. [ABSTRACT FROM AUTHOR]

In this paper, we consider n -dimensional nilpotent finite-dimensional evolution algebras with non-maximal index of nilpotency. We give complete invariants of these algebras in order to classify them. Then, we describe their associative enveloping algebras. We also give a positive answer to the question whether a derivation of this nilpotent evolution algebra is possible to be extended to its enveloping algebra. [ABSTRACT FROM AUTHOR]

Let α = (A g , α g) g ∈ be a group-type partial action of a connected groupoid on a ring A = ⊕ z ∈ 0 A z and B : = A ⋆ α be the corresponding partial skew groupoid ring. In the first part of this paper, we investigate the relation of several ring theoretic properties between A and B. For the second part, using that every Leavitt path algebra is isomorphic to a partial skew groupoid ring obtained from a partial groupoid action λ , we characterize when λ is group-type. In such a case, we obtain ring theoretic properties of Leavitt path algebras from the results on general partial skew groupoid rings. Several examples that illustrate the results on Leavitt path algebras are presented. [ABSTRACT FROM AUTHOR]

A commutative ring R has the unique decomposition into ideals (UDI) property if, for any R -module that decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism classes of the indecomposable ideals. In [P. Goeters and B. Olberding, Unique decomposition into ideals for Noetherian domains, J. Pure Appl. Algebra 165 (2001) 169–182], the UDI property has been characterized for Noetherian integral domains. In this paper, we aim to study the UDI-like property for strong Mori domains; domains satisfying the ascending chain condition on w -ideals. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of a product structure on a 3-Bihom-Lie algebra, which is a Nijenhuis operator with some conditions. We prove that a 3-Bihom-Lie algebra has a product structure if and only if it is the direct sum of two vector spaces which are also Bihom-subalgebras. Then we give four special conditions under each of which a 3-Bihom-Lie algebra has a special decomposition. Similarly, we introduce a complex structure on a 3-Bihom-Lie algebra and there are also four types of special complex structures. Finally, we establish the relation between a complex structure and a product structure. [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]

We create an invariant of virtual Y -oriented trivalent spatial graphs using colorings by virtual Niebrzydowski algebras. This paper generalizes the color invariants using virtual tribrackets and Niebrzydowski algebras by Nelson–Pico, and Graves-Nelson-T. We computed all tribrackets, Niebrzydowski algebras and virtual Niebrzydowski algebras of orders 3 and 4, and provide generative code for all data sets. [ABSTRACT FROM AUTHOR]

In this paper, we introduced two classes of elements in the enveloping algebra U (g l (n)) : the double Young–Capelli bitableaux  [ S | T ] and the central Schur elements  S λ (n) , that act in a remarkable way on the highest weight vectors of irreducible Schur modules. Any element S λ (n) is the sum of all double Young–Capelli bitableaux [ S | S ] , S row (strictly) increasing Young tableaux of shape λ ̃. The Schur elements S λ (n) are proved to be the preimages — with respect to the Harish-Chandra isomorphism — of the shifted Schur polynomials  s λ | n ∗ ∈ Λ ∗ (n). Hence, the Schur elements are the same as the Okounkov quantum immanants, recently described by the present authors as linear combinations of Capelli immanants. This new presentation of Schur elements/quantum immanants does not involve the irreducible characters of symmetric groups. The Capelli elements H k (n) are column Schur elements and the Nazarov elements I k (n) are row Schur elements. The duality in ζ (n) follows from a combinatorial description of the eigenvalues of the H k (n) on irreducible modules that is dual (in the sense of shapes/partitions) to the combinatorial description of the eigenvalues of the I k (n). The passage n → ∞ for the algebras ζ (n) is obtained both as direct and inverse limit in the category of filtered algebras, via the Olshanski decomposition/projection. [ABSTRACT FROM AUTHOR]

In [S. Das, M. K. Sen and S. K. Maity, Leavitt path algebras for power graphs of finite groups, J. Algebra Appl., doi:10.1142/S0219498822502097], we studied several algebraic properties of Leavitt path algebra of the directed power graph ⃗ (G) and also of the directed punctured power graph ⃗ * (G) of a finite group G. In this paper, we compute the Grothendieck group of purely infinite simple Leavitt path algebra of the directed punctured power graph of a finite group. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a kind of double centralizer property for general linear supergroups. For the super space V = m | n over an algebraically closed field whose characteristic is not equal to 2 , we consider its ℤ 2 -homogeneous one-dimensional extension V ̲ = V ⊕ v , and the natural action of the supergroup G ̃ : = GL (V) × G m on V ̲. Then we have the tensor product supermodule ( V ̲ ⊗ r , ρ r ) of G ̃. We present a kind of generalized Schur–Sergeev duality which is said that the Schur superalgebras S ′ (m | , n , r) of G ̃ and the so-called weak degenerate double Hecke algebra ℋ ̲ r are double centralizers. The weak degenerate double Hecke algebra is an infinite-dimensional algebra, which has a natural representation on the tensor product space. This notion comes from [B. Shu, Y. Xue and Y. Yao, On enhanced reductive groups (I): Parabolic Schur algebras and the dualities related to degenerate double Hecke algebras, preprint (2013), arXiv:2005.13152 [Math. RT]], with a little modification. [ABSTRACT FROM AUTHOR]

In this paper, we consider the infinitesimal freeness in the operator-valued framework, and we show that the operator-valued infinitesimal (OVI) free independence is equivalent to the operator-valued free independence over an algebra of 2 × 2 upper triangular matrices. We introduce the notion of OVI cumulants and investigate its properties, and we then deduce that the OVI freeness is equivalent to the vanishing of our mixed cumulants. Moreover, we derive the formula for obtaining the free additive and multiplicative convolutions within the realm of OVI freeness. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the structure of irreducible modules in the variety ℳ of commutative power-associative nilalgebras of nilindex ≤ 4. If A ∈ ℳ with dimension at most 5, then we prove that A 2 is contained in the annihilator of every irreducible A -module in the variety ℳ. Also, we consider the enveloping algebra of an algebra A in the variety ℳ and we obtain a new example of a commutative power-associative non-nilpotent nilalgebra of dimension 9. [ABSTRACT FROM AUTHOR]

Multicomponent commutations relations (MCRs) describe plektons, i.e. multicomponent quantum systems with a generalized statistics. In such systems, exchange of quasiparticles is governed by a unitary matrix Q (x 1 , x 2) that depends on the position of quasiparticles. For such an exchange to be possible, the matrix must satisfy several conditions, including the functional Yang–Baxter equation. The aim of the paper is to give an appropriate definition of a quasi-free state on an MCR algebra, and construct such states on a class of MCR algebras. We observe a significant difference between the classical setting for bosons and fermions and the setting of MCR algebras. We show that the developed theory is applicable to systems that contain quasiparticles of opposite type. An example of such a system is a two-component system in which two quasiparticles, under exchange, change their respective types to the opposite ones (1 ↦ 2 , 2 ↦ 1). Fusion of quasiparticles means intuitively putting several quasiparticles in an infinitely small box and identifying the statistical behavior of the box. By carrying out fusion of an odd number of particles from the two-component system as described above, we obtain further examples of quantum systems to which the developed theory is applicable. [ABSTRACT FROM AUTHOR]

In this paper, we provide a novel generalization of quantum orthogonal polynomials from ℛ (p , q) -deformed quantum algebras introduced in earlier works. We construct related quantum Jacobi polynomials and their probability distribution, factorial moments, recurrence relation, and governing difference equation. Surprisingly, these polynomials obey non-conventional recurrence relations. Particular cases of generalized quantum little Legendre, little Laguerre, Laguerre, Bessel, Rogers–Szegö, Stieltjes–Wigert and Kemp binomial polynomials are derived and discussed. [ABSTRACT FROM AUTHOR]

Let H ≤ K be subgroups of a finite group G , then H is called strongly closed in K with respect to G if H g ∩ K ≤ H for every g ∈ G , and in particular, H is simply called strongly closed in G if H is strongly closed in N G (H) with respect to G. Let H be a subgroup of a finite group G , then H is called Hall s -semiembedded in G if H is a Hall subgroup of 〈 H , P 〉 for every P ∈ Syl p (G) , where (| H | , p) = 1. In this paper, we obtain some criteria for p -nilpotency of a finite group and extend some known results concerning strongly closed and Hall s -semiembedded subgroups. In particular, we generalize some main results of Guo and Li [Hall s -semiembedded subgroups and p -nilpotency of finite groups, Southeast Asian Bull. Math. 42(3) (2018) 367–374] and Kong [New characterizations of p -nilpotency of finite groups, J. Algebra Appl. 20(11) (2021) Article ID: 2150215, 6 pp.]. [ABSTRACT FROM AUTHOR]

Let J (R) denote the Jacobson radical of a commutative ring R. In [H. A. Khashan and A. B. Bani-Ata, J -ideals of commutative rings, Int. Electron. J. Algebra 29 (2021) 148–164], the notion of J-ideals was introduced. If N (R) denotes the prime radical of a commutative ring then in [U. Tekir, S. Koc and K. H. Oral, n -Ideals of commutative rings, Filomat 31(10) (2017) 2933–2941], the notion of an N ideal of a commutative ring was introduced and studied. In this note, we show that these results are special cases of a more general situation. We define ρ -ideals for a special radical ρ and prove that most of the results of the above-mentioned papers are satisfied for non-commutative rings as a special case. [ABSTRACT FROM AUTHOR]

In [Finite solvable groups with exactly two nonlinear non-faithful irreducible characters, J. Algebra Appl. 18(5) (2019) 1950091], the first author studied directly indecomposable, solvable groups with exactly two nonlinear non-faithful irreducible characters. In this paper, we study the decomposable case and give a classification of decomposable groups possessing exactly two nonlinear non-faithful irreducible characters. In particular, nilpotent groups with this property are also classified. [ABSTRACT FROM AUTHOR]

Let (R ,) be a Noetherian local ring and M a finitely generated R -module. In this paper, we study the polynomial type and the sequential polynomial type of the idealization R ⋉ M of R and M introduced by Cuong [On the least degree of polynomials bounding above the differences between lengths and multiplicities of certain systems of parameters in local rings, Nagoya Math. J. 125 (1992) 105–114] and Nhan-Dung-Chau [A measure for non-sequential Cohen Macaulayness of finitely generated modules, J. Algebra 468 (2016) 275–295], respectively. In addition, we give a uniform bound for the reducibility index of parameter ideals of idealizations whose polynomial type and sequential polynomial type are at most one. [ABSTRACT FROM AUTHOR]

In this paper, we provide a diagrammatic approach to study the branching rules for cell modules on a sequence of walled Brauer algebras. This approach also allows us to calculate the structure constants of multiplication over the Grothendieck ring of the sequence. [ABSTRACT FROM AUTHOR]