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Group-twisted Alexander-Whitney and Eilenberg-Zilber mapsMay 16 2019We define group-twisted Alexander-Whitney and Eilenberg-Zilber maps for converting between bimodule resolutions of skew group algebras. These algebras are the natural semidirect products recording actions of finite groups by automorphisms. The group-twisted ... More

Finite quotients of powers of an elliptic curveMay 16 2019Let $E$ be an elliptic curve. When the symmetric group $\Sigma_{g+1}$ of order $(g+1)!$ acts on $E^{g+1}$ in the natural way, the subgroup $E_0^{g+1}$, consisting of those $(g+1)$-tuples whose coordinates sum to zero, is stable under the action of $\Sigma_{g+1}$. ... More

From the potential to the first Hochschild cohomology group of a cluster tilted algebraMay 16 2019The objective of this paper is to give a concrete interpretation of the dimension of the first Hochschild cohomology space of a cyclically oriented or tame cluster tilted algebra in terms of a numerical invariant arising from the potential.

Factorizations of skew bracesMay 14 2019We introduce strong left ideals of skew braces and prove that they produce non-trivial decomposition of set-theoretic solutions of the Yang-Baxter equation. We study factorization of skew left braces through strong left ideals and we prove analogs of ... More

Double Magma associated with Ward and double Ward quasigroupsMay 14 2019We describe types of double magma associated with Ward quasigroups, double Ward quasigroups, their duals and the groups they generate. Ward quasigroup double magma and unipotent, right modular, left unital double magma are proved to be improper. Necessary ... More

Degenerations of nilpotent algebrasMay 14 2019We give a complete description of degenerations of $3$-dimensional nilpotent algebras, $4$-dimensional nilpotent commutative algebras and $5$-dimensional nilpotent anticommutative algebras over $ \mathbb C$. In particular, we correct several mistakes ... More

A geometric $q$-character formula for snake modulesMay 13 2019Let $\mathscr{C}$ be the category of finite dimensional modules over the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ of a simple complex Lie algebra ${\mathfrak{g}}$. Let $\mathscr{C}^-$ be the subcategory introduced by Hernandez and Leclerc. ... More

Orthogonal tensor decomposition and orbit closures from a linear algebraic perspectiveMay 13 2019We study orthogonal decompositions of symmetric and ordinary tensors using methods from linear algebra. For the field of real numbers we show that the sets of decomposable tensors can be defined be equations of degree 2. This gives a new proof of some ... More

A Batalin-Vilkovisky structure on the complete cohomology ring of a Frobenius algebraMay 13 2019We study the exsistence of a Batalin-Vilkovisky differential on the complete cohomology ring of a Frobenius algebra. We construct a Batalin-Vilkovisky differential on the complete cohomology ring in the case of Frobenius algebras with diagonalizable Nakayama ... More

On the minima of positive definite binary hamiltonian formsMay 13 2019Let $A$ be a definite quaternion algebra over $\mathbb Q$, with discriminant $D_A$, and $O$ a maximal order of $A$. We show that the minimum of the positive definite hamiltonian binary forms over $O$ with discrimiminant $-1$ is $\sqrt{D_A}$. When the ... More

Wajsberg algebras of order n, n<=9May 12 2019In this paper, we describe all finite Wajsberg algebras of order n<=9.

Clifford deformations of Koszul Frobenius algebras and noncommutative quadricsMay 12 2019Let $E$ be a Koszul Frobenius algebra. A Clifford deformation of $E$ is a finite dimensional $\mathbb Z_2$-graded algebra $E(\theta)$, which corresponds to a noncommutative quadric hypersurface $E^!/(z)$, for some central regular element $z\in E^!_2$. ... More

$3$-BiHom-Lie superalgebras induced by BiHom-Lie superalgebrasMay 11 2019The purpose of this paper is to study the relationships between a BiHom-Lie superalgebras and its induced 3-BiHom-Lie superalgebras. We introduce the notion of $(\alpha^s,\beta^r)$-derivation, $(\alpha^s,\beta^r)$-quasiderivation and generalized $(\alpha^s,\beta^r)$-derivation ... More

Simple $\mathbb{Z}$-graded domains of Gelfand-Kirillov dimension twoMay 10 2019Let $k$ be an algebraically closed field and $A$ a $\mathbb{Z}$-graded finitely generated simple $k$-algebra which is a domain of Gelfand-Kirillov dimension 2. We show that the category of $\mathbb{Z}$-graded right $A$-modules is equivalent to the category ... More

Pointed Hopf algebras over non abelian groups with decomposable braidings, IMay 10 2019We describe all finite-dimensional pointed Hopf algebras whose infinitesimal braiding is a fixed Yetter-Drinfeld module decomposed as the sum of two simple objects: a point and the one of transpositions of the symmetric group in three letters. We give ... More

Structure of a class of Lie conformal algebras of Block typeMay 10 2019Let $p$ be a nonzero complex number. Recently, a class of infinite rank Lie conformal algebras $\mathfrak{B}(p)$ was introduced in [13]. In this paper, we study the structure theory of this class of Lie conformal algebras. Specifically, we completely ... More

The Nullstellensatz for supersymmetric polynomialsMay 10 2019In this paper we prove a Nullstellensatz for supersymmetric polynomials. This gives a bijection between radical ideals and superalgebraic sets. These are algebraic sets which are invariant under the Weyl groupoid of Sergeev and Veselov, \cite{SV2}. Note ... More

Dimensions of semi-simple matrix algebrasMay 09 2019For $n \geq 225$ we show that every integer of the form $n + 2m$ such that $0 \leq 2m \leq n^{2} - \frac{9}{2} n \sqrt{n}$ is the dimension of a connected semi-simple subalgebra of $\mathrm{M}_{n}(k)$, that is, a subalgebra isomorphic to a direct sum ... More

Decomposition algebras and axial algebrasMay 09 2019We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras: (1) They separate fusion laws from specific values in a field, thereby allowing ... More

Representations and cohomology of a family of finite supergroup schemesMay 08 2019We examine the cohomology and representation theory of a family of finite supergroup schemes of the form $(\mathbb G_a^-\times \mathbb G_a^-)\rtimes (\mathbb G_{a(r)}\times (\mathbb Z/p)^s)$. In particular, we show that a certain relation holds in the ... More

Higher Spherical AlgebrasMay 08 2019We introduce and study higher spherical algebras, an exotic family of finite-dimensional algebras over an algebraically closed field. We prove that every such an algebra is derived equivalent to a higher tetrahedral algebra studied in [7], an hence that ... More

Koszul Algebras and Flow LatticesMay 08 2019We provide a homological algebraic realization of the lattices of integer cuts and integer flows of graphs. To a finite 2-edge-connected graph $\Gamma$ with a spanning tree $T$, we associate a finite dimensional Koszul algebra $A_{\Gamma,T}$. Under the ... More

The ascent-descent property for $2$-term silting complexesMay 08 2019We will prove that over commutative rings the silting property of $2$-term complexes induced by morphisms between projective modules is preserved and reflected by faithfully flat extensions.

Bialgebraic approach to rack cohomologyMay 07 2019We interpret the complexes defining rack cohomology in terms of a certain differential graded bialgebra. This yields elementary algebraic proofs of old and new structural results for this cohomology theory. For instance, we exhibit two explicit homotopies ... More

Twisted superpotentials of 3-dimensional quadratic AS-regular algebrasMay 07 2019Let $k$ be an algebraically closed field of characteristic $0$ and $A$ a graded $k$-algebra finitely generated in degree $1$. In this paper, for $3$-dimensional quadratic AS-regular algebras except for Type EC, we give a complete list of twisted superpotentials ... More

Examples of finite-dimensional pointed Hopf algebras in positive characteristicMay 07 2019We present new examples of finite-dimensional Nichols algebras over fields of positive characteristic. The corresponding braided vector spaces are not of diagonal type, admit a realization as Yetter-Drinfeld modules over finite abelian groups and are ... More

Extending valuations to the field of rational functions using pseudo-monotone sequencesMay 07 2019For a valuation domain $V$ of rank one and quotient field $K$, Ostrowski introduced in 1935 the notion of pseudo-convergent sequence and proved his Fundalmentalsatz, which describes all the possible rank one extensions of $V$ to $K(X)$. In this paper, ... More

On pure derived and pure singularity categoriesMay 07 2019Firstly, we compare the bounded derived categories with respect to the pure-exact and the usual exact structures, and describe bounded derived category by pure-projective modules, under a fairly strong assumption on the ring. Then, we study Verdier quotient ... More

Self-Invariant Maximal Subfields and Their Connexion with Some Conjectures in Division RingsMay 06 2019Let D be a division algebra with center F. A maximal subfield of D is defined to be a field K such that CD(K) = K; that is, K is its own centralizer in D. A maximal subfield K is said to be self-invariant if it normalises by itself, i.e. ND*(K)= K: This ... More

Torsion-type $q$-deformed Heisenberg algebra and its Lie polynomialsMay 06 2019Given a scalar parameter $q$, the $q$-deformed Heisenberg algebra $\mathcal{H}(q)$ is the unital associative algebra with two generators $A,B$ that satisfy the $q$-deformed commutation relation $AB-qBA= I$, where $I$ is the multiplicative identity. For ... More

Supertropical Monoids II: Lifts, Transmissions, and EqualizersMay 06 2019The category $\operatorname{STROP}$ of commutative semirings, whose morphisms are transmissions, is a full and reflective subcategory of the category $\operatorname{STROP}_m$ of supertropical monoids. Equivalence relations on supertropical monoids are ... More

Three results for tau-rigid modulesMay 06 2019$\tau$-rigid modules are essential in the $\tau$-tilting theory introduced by Adachi, Iyama and Reiten. In this paper, we give equivalent conditions for Iwanaga-Gorenstein algebras with self-injective dimension at most one in terms of $\tau$-rigid modules. ... More

Finitely generated abelian groups of unitsMay 06 2019In 1960 Fuchs posed the problem of characterizing the groups which are the groups of units of commutative rings. In the following years, some partial answers have been given to this question in particular cases. In this paper we address Fuchs' question ... More

2-generation of simple Lie algebras and free dense subgroups of algebraic groupsMay 06 2019We construct generating pairs of simple Lie algebras in characteristic zero. We apply this construction to exhibit infinite series of 2-generator Zariski dense subgroups that are free of rank 2 of the simple algebraic groups SL(n, C), Sp(n, C), G_2(C). ... More

2-generation of simple Lie algebras and free dense subgroups of algebraic groupsMay 06 2019May 14 2019We construct generating pairs of simple Lie algebras in characteristic zero. We apply this construction to exhibit infinite series of 2-generator Zariski dense subgroups that are free of rank 2 of the simple algebraic groups SL(n, C), Sp(n, C), G_2(C). ... More

The regular representation of $U_v(\mathfrak{gl}_{m|n})$May 06 2019Using quantum differential operators, we construct a super representation of $U_v(\mathfrak{gl}_{m|n})$ on a certain polynomial superalgebra. We then extend the representation to its formal power series algebra which contains a $U_v(\mathfrak{gl}_{m|n})$-submodule ... More

Complex fuzzy Lie AlgebrasMay 05 2019A complex fuzzy Lie algebra is a fuzzy Lie algebra whose membership function takes values in the unit circle in the complex plane. In this paper, we deine the complex fuzzy Lie subalgebras and complex fuzzy ideals of Lie algebras. Then, we investigate ... More

Representations and Modules of Rota-Baxter AlgebrasMay 04 2019We give a broad study of representation and module theory of Rota-Baxter algebras, motivated by Rota-Baxter matrix representations in the renormalization of quantum field theory and by geometric connections. Regular-singular decompositions of Rota-Baxter ... More

A Generalization of Reflexive RingsMay 04 2019In this paper, we introduce a class of rings which is a generalization of reflexive rings and $J$-reversible rings. Let $R$ be a ring with identity and $J(R)$ denote the Jacobson radical of $R$. A ring $R$ is called {\it $J$-reflexive} if for any $a$, ... More

Centrally Stable AlgebrasMay 04 2019We define an algebra $A$ to be centrally stable if, for every epimorhism $\varphi$ from $A$ to another algebra $B$, the center $Z(B)$ of $B$ is equal to $\varphi(Z(A))$, the image of the center of $A$. After providing some examples and basic observations, ... More

Graded torsion-free ${\mathfrak{sl}_2(\mathbb{C})}$-modules of rank 2May 03 2019In this paper we explore the possibility of endowing simple infinite-dimensional ${\mathfrak{sl}_2(\mathbb{C})}$-modules by the structure of the graded module. The gradings on finite-dimensional simple module over simple Lie algebras has been studied ... More

Noncommutative cross-ratio and Schwarz derivativeMay 03 2019We present here a theory of noncommutative cross-ratio, Schwarz derivative and their connections and relations to the operator cross-ratio. We apply the theory to "noncommutative elementary geometry" and relate it to noncommutative integrable systems. ... More

Noncommutative cross-ratio and Schwarz derivativeMay 03 2019May 07 2019We present here a theory of noncommutative cross-ratio, Schwarz derivative and their connections and relations to the operator cross-ratio. We apply the theory to "noncommutative elementary geometry" and relate it to noncommutative integrable systems. ... More

New Perfect Nonlinear Functions and Their SemifieldsMay 03 2019In this paper, two new classes of perfect nonlinear functions over $\mathbb{F}_{p^{2m}}$ are proposed, where $p$ is an odd prime. Furthermore, we investigate the nucleus of the corresponding semifields of these functions and show that the semifields are ... More

New Perfect Nonlinear Functions and Their SemifieldsMay 03 2019May 08 2019In this paper, two new classes of perfect nonlinear functions over $\mathbb{F}_{p^{2m}}$ are proposed, where $p$ is an odd prime. Furthermore, we investigate the nucleus of the corresponding semifields of these functions and show that the semifields are ... More

A necessary condition for zero divisors in complex group algebra of torsion-free groupsMay 02 2019We find a necessary condition for zero divisors in complex group algebras of torsion-free groups.

Learning Algebraic Structures: Preliminary InvestigationsMay 02 2019We employ techniques of machine-learning, exemplified by support vector machines and neural classifiers, to initiate the study of whether AI can "learn" algebraic structures. Using finite groups and finite rings as a concrete playground, we find that ... More

On Leibniz superalgebras which even part is sl_2May 02 2019This article deals with a Leibniz superalgebras $L=L_0\oplus L_1,$ whose even part is a simple Lie algebra $\mathfrak{sl}_2$. We describe all such Leibniz superalgebras when odd part is an irreducible Leibniz bi-module on $\mathfrak{sl}_2 $. We show that ... More

Trivial Schur indices for noncommutative reality-based algebras with exactly two nonreal basis elementsMay 01 2019This article discusses the splitting fields of noncommutative reality-based algebras with positive degree map. We show that if the standard basis has only one nonreal pair, then the minimal field of definition for the RBA is always a splitting field. ... More

Higher Arity Self-Distributive Operations in Cascades and their CohomologyMay 01 2019We investigate constructions and relations of higher arity self-distributive operations and their cohomology. We study the categories of mutually distributive structures both in the binary and ternary settings and their connections through functors. This ... More

Geometry of central extensions of nilpotent Lie algebrasMay 01 2019We obtain a recurrent and monotone method for constructing and classifying nilpotent Lie algebras by means of successive central extensions. It consists in calculating the second cohomology of an extendable nilpotent Lie algebra with the subsequent study ... More

Harmonic cubic homogeneous polynomials such that the norm-squared of the Hessian is a multiple of the Euclidean quadratic formApr 30 2019May 06 2019There is considered the problem of describing up to linear conformal equivalence those harmonic cubic homogeneous polynomials for which the squared-norm of the Hessian is a nonzero multiple of the quadratic form defining the Euclidean metric. Solutions ... More

Harmonic cubic homogeneous polynomials such that the norm-squared of the Hessian is a multiple of the Euclidean quadratic formApr 30 2019There is considered the problem of describing up to linear conformal equivalence those harmonic cubic homogeneous polynomials for which the squared-norm of the Hessian is a nonzero multiple of the quadratic form defining the Euclidean metric. Solutions ... More

Ring Constructions and Generation of the Unbounded Derived Module CategoryApr 30 2019Given the unbounded derived module category of a ring $A$, we consider the triangulated subcategory closed under arbitrary coproducts generated by injective modules. Similarly we also look at the triangulated subcategory closed under arbitrary products ... More

Hopf algebras with enough quotientsApr 30 2019A family of algebra maps $H\to A_i$ whose common domain is a Hopf algebra is said to be jointly inner faithful if it does not factor simultaneously through a proper Hopf quotient of $H$. We show that tensor and free products of jointly inner faithful ... More

Finite-dimensional Leibniz algebra representations of $\mathfrak{sl}_2$Apr 30 2019All finite-dimensional Leibniz algebra bimodules of a Lie algebra $\mathfrak{sl}_2$ over a field of characteristic zero are described.

Hopf modules, Frobenius functors and (one-sided) Hopf algebrasApr 30 2019We investigate the property of being Frobenius for some functors strictly related with Hopf modules over a bialgebra and how this property reflects on the latter. In particular, we characterize one-sided Hopf algebras with anti-(co)multiplicative one-sided ... More

On idempotents of commutative ringsApr 29 2019In the present work, a formula is provided for determining the idempotent elements of a commutative ring R from those of the quotient ring R/N, where N is in most cases a nilpotent ideal of R. As an application of this formula, idempotent elements of ... More

Algorithmic approach to strong consistency analysis of finite difference approximations to PDE systemsApr 29 2019For a wide class of polynomially nonlinear systems of partial differential equations we suggest an algorithmic approach to the s(trong)-consistency analysis of their finite difference approximations on Cartesian grids. First we apply the differential ... More

The automorphisms of generalized cyclic Azumaya algebrasApr 29 2019We define a nonassociative generalization of cyclic Azumaya algebras employing skew polynomial rings $D[t;\sigma]$, where $D$ is an Azumaya algebra of constant rank with center $C$ and $\sigma$ an automorphism of $D$, such that $\sigma|_{C}$ has finite ... More

Virtual representation motivesApr 29 2019Principal $GL_n$-bundles (aka vector bundles) are locally trivial in the Zariski topology, whereas principal $PGL_n$-bundles (aka Azumaya algebras) are not, to the delight of every non-commutative algebraist. Still, this makes the calculation of motives ... More

Cuspidal modules for the derivation Lie algebra over a rational quantum torusApr 29 2019Let $\mathbb C_Q$ denote a rational quantum torus with $d$ variables, and $\mathcal Z$ be the centre of $\mathbb C_Q$. In this paper we give a explicit description of the structure of the cuspidal modules for the derivation Lie algebra $\mathcal D$ over ... More

Density of $g$-vector cones from triangulated surfacesApr 29 2019We study $g$-vector cones associated with clusters of cluster algebras defined from a marked surface $(S,M)$ of rank $n$. We determine the closure of the union of $g$-vector cones associated with all clusters. It is equal to $\mathbb{R}^n$ except for ... More

Density of $g$-vector cones from triangulated surfacesApr 29 2019May 14 2019We study $g$-vector cones associated with clusters of cluster algebras defined from a marked surface $(S,M)$ of rank $n$. We determine the closure of the union of $g$-vector cones associated with all clusters. It is equal to $\mathbb{R}^n$ except for ... More

BGG category for the quantum Schrödinger algebraApr 29 2019In this paper, we study the BGG category $\mathcal{O}$ for the quantum Schr{\"o}dinger algebra $U_q(\mathfrak{s})$, where $q$ is a nonzero complex number which is not a root of unity. If the central charge $\dot z\neq 0$, using the module $B_{\dot z}$ ... More

Deformations of Loday-type algebras and their morphismsApr 28 2019We study a formal deformation of multiplications in an operad. This closely resemble Gerstenhaber's deformation theory for associative algebras. However, this is applicable to various Loday-type algebras and to their twisted analogues. We explicitly describe ... More

Deformations of Loday-type algebras and their morphismsApr 28 2019May 03 2019We study a formal deformation of multiplications in an operad. This closely resemble Gerstenhaber's deformation theory for associative algebras. However, this is applicable to various Loday-type algebras and to their twisted analogues. We explicitly describe ... More

Combinatorics and structure of Hecke-Kiselman algebrasApr 27 2019Hecke-Kiselman monoids $\textrm{HK}_{\Theta}$ and their algebras $K[\textrm{HK}_{\Theta}]$, over a field $K$, associated to finite oriented graphs $\Theta$ are studied. In the case $\Theta $ is a cycle of length $n\geqslant 3$, a hierarchy of certain ... More

Mutation of type $D$ friezesApr 26 2019In this article we study mutation of friezes of type $D$. We provide a combinatorial formula for the entries in a frieze after mutation. The two main ingredients in the proof include a certain transformation of a type $D$ frieze into a sub pattern of ... More

From Trigroups To Leibniz 3-AlgebrasApr 26 2019In this paper, we study the category of trigroups as a generalization of the notion of digroup [4] and analyze their relationship with 3-racks [1] and Leibniz 3-algebras [6]. Trigroups are essentially associative trioids in which there are bar-units and ... More

Set-theoretic solutions of the Yang--Baxter equation, associated quadratic algebras and the minimality conditionApr 26 2019Given a finite non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation and a field $K$, the structure $K$-algebra of $(X,r)$ is $A=A(K,X,r)=K\langle X\mid xy=uv \mbox{ whenever }r(x,y)=(u,v)\rangle$. Note that $A=\oplus_{n\geq 0} A_n$ ... More

Stratifying systems through $τ$-tilting theoryApr 26 2019In this paper we study the stratifying systems, introduced by K. Erdmann and C. Saenz in \cite{Erdmann2003}, by using the $\tau$-tilting theory introduced by T. Adachi, O. Iyama and I. Reiten in \cite{AIR}. We give a constructive proof that every non-zero ... More

A note on unitary Cayley graphs of matrix algebrasApr 26 2019In [1], the authors claim to have found the unitary Cayley graph $Cay(M_{n}(F),GL_{n}(F))$ of matrix algebras over finite field $F$ is strongly regular only when $n=2$. But they have only cited two special cases to prove it, namely when $n = 2$ and $3$, ... More

A note on unitary Cayley graphs of matrix algebrasApr 26 2019Apr 29 2019In [1], the authors claim to have found the unitary Cayley graph $Cay(M_{n}(F),GL_{n}(F))$ of matrix algebras over finite field $F$ is strongly regular only when $n=2$. But they have only cited two special cases to prove it, namely when $n = 2$ and $3$, ... More

No dialgebra has Gelfand-Kirillov dimension strictly between 1 and 2Apr 26 2019The Gelfand-Kirillov dimension measures the asymptotic growth rate of algebras. For every associative dialgebra $\mathcal{D}$, the quotient $\mathcal{A}_\mathcal{D}:=\mathcal{D}/\mathsf{Id}(S)$, where $\mathsf{Id}(S)$ is the ideal of $\mathcal{D}$ generated ... More

Generalized Cline's Formula for G-Drazin InverseApr 26 2019Let $R$ be an associative ring with an identity and suppose that $a,b,c,d \in R$ satisfy $bdb = bac,dbd = acd$. If $ac$ has generalized Drazin ( respectively, pseudo Drazin, Drazin) inverse, we prove that $bd$ has generalized Drazin (respectively, pseudo ... More

Retractability of solutions to the Yang-Baxter equation and $p$-nilpotency of skew bracesApr 26 2019Using Bieberbach groups we study multipermutation involutive solutions to the Yang-Baxter equation. We find a linear representation of structure groups of involutive solutions; this representation is then used to study the unique product property in such ... More

An application of a theorem of Sheila Brenner for Hochschild extension algebras of a truncated quiver algebraApr 26 2019Let $A$ be a truncated quiver algebra over an algebraically closed field such that any oriented cycle in the ordinary quiver of $A$ is zero in $A$. We give the number of the indecomposable direct summands of the middle term of an almost split sequence ... More

Constructing minimal telescopers for rational functions in three discrete variablesApr 25 2019We present a new algorithm for constructing minimal telescopers for rational functions in three discrete variables. This is the first discrete reduction-based algorithm that goes beyond the bivariate case. The termination of the algorithm is guaranteed ... More

Constructing minimal telescopers for rational functions in three discrete variablesApr 25 2019May 03 2019We present a new algorithm for constructing minimal telescopers for rational functions in three discrete variables. This is the first discrete reduction-based algorithm that goes beyond the bivariate case. The termination of the algorithm is guaranteed ... More

Fundamental building blocks for unitary matrices and quantum logic gatesApr 25 2019Apr 28 2019A unitary matrix is shown to be a product of certain {\em basic} unitary matrices and the product is unique up to order. A basic unitary matrix itself is a product of {\em minimal basic} matrices. A unitary $n\times n$ matrix can be expressed as a product ... More

Fundamental building blocks for unitary matrices and quantum logic gatesApr 25 2019A unitary matrix is shown to be a product of certain {\em basic} unitary matrices and the product is unique up to order. A basic unitary matrix itself is a product of {\em minimal basic} matrices. A unitary $n\times n$ matrix can be expressed as a product ... More

Three modules of simple 3-Lie algebra $A_ω^δ$Apr 25 2019We construct three infinite dimensional intermediate series reducible modules over the simple infinite dimensional canonical Nambu $3$-Lie algebra $A_{\omega}^{\delta}$, and further investigate their structures. We also construct three other infinite ... More

On the structure of split regular Hom-Lie Rinehart algebrasApr 25 2019The aim of this paper is to study the structures of split regular Hom-Lie Rinehart algebras. Let $(L,A)$ be a split regular Hom-Lie Rinehart algebra. We first show that $L$ is of the form $L=U+\sum_{[\gamma]\in\Gamma/\thicksim}I_{[\gamma]}$ with $U$ a ... More

The ring $\mathrm{M}_{8k+4}(\mathbb{Z}_2)$ is nil-clean of index fourApr 24 2019We show that the direct sum of an odd number of matrices $$C=\left(\begin{array}{cccc} 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0\\ 0&0&1&1 \end{array}\right)$$ cannot be a sum $P+Q$ of matrices over $\mathbb{F}_2$ satisfying $P^2=P$ and $Q^3=O$.

Higher Deformations of Lie Algebra Representations IIApr 24 2019Steinberg's tensor product theorem shows that for semisimple algebraic groups the study of irreducible representations of higher Frobenius kernels reduces to the study of irreducible representations of the first Frobenius kernel. In the preceding paper ... More

Division algebras graded by a finite groupApr 24 2019Let $k$ be a field containing an algebraically closed field of characteristic zero. If $G$ is a finite group and $D$ is a division algebra over $k$, finite dimensional over its center, we can associate to a faithful $G$-grading on $D$ a normal abelian ... More

Heun algebras of Lie typeApr 24 2019We introduce Heun algebras of Lie type. They are obtained from bispectral pairs associated to simple or solvable Lie algebras of dimension three or four. For $\mathfrak{su}(2)$, this leads to the Heun-Krawtchouk algebra. The corresponding Heun-Krawtchouk ... More

$A_\infty$-Minimal Model on Differential Graded AlgebrasApr 23 2019For a formal differential graded algebra, if extended by an odd degree element, we prove that the extended algebra has an $A_\infty$-minimal model with only $m_2$ and $m_3$ non-trivial. As an application, the $A_\infty$-algebras constructed by Tsai, Tseng ... More

A Note on Direct Products, Subreducts and Subvarieties of PBZ*--latticesApr 22 2019PBZ*--lattices are bounded lattice--ordered structures arising in the study of quantum logics, which include orthomodular lattices, as well as antiortholattices. Antiortholattices turn out not only to be directly irreducible, but also to have directly ... More

The roots of exceptional modular Lie superalgebras with Cartan matrixApr 21 2019For each of the exceptional Lie superalgebras with indecomposable Cartan matrix, we give the explicit list of its roots of and the corresponding Chevalley basis for one of the inequivalent Cartan matrices, the one corresponding to the greatest number ... More

Alexandroff Topology of Algebras over an Integral DomainApr 20 2019Let $S$ be an integral domain with field of fractions $F$ and let $A$ be an $F$-algebra. An $S$-subalgebra $R$ of $A$ is called $S$-nice if $R$ is lying over $S$ and the localization of $R$ with respect to $S \setminus \{ 0 \}$ is $A$. Let $\mathbb S$ ... More

Quasicomplemented residuated latticesApr 20 2019In this paper, the class of quasicomplemented residuated lattices is introduced and investigated, as a subclass of residuated lattices in which any prime filter not containing any dense element is a minimal prime filter. The notion of disjunctive residuated ... More

The Moore-Penrose inverses of split quaternionsApr 20 2019In this paper, we find the roots of lightlike quaternions. By introducing the concept of the Moore-Penrose inverse in split quaternions, we solve the linear equations $axb=d$, $xa=bx$ and $xa=b\bar{x}$. Also we obtain necessary and sufficient conditions ... More

Intersection property and interaction decompositionApr 18 2019The decomposition of interactions, or interaction decomposition, is a hierarchical decomposition of the factor spaces into direct sums of interaction spaces. We consider general arrangements of vector subspaces and want to know when one can still build ... More

Bi-skew braces and Hopf Galois structuresApr 18 2019We define a bi-skew brace to be a set $G$ with two group operations $\star$ and $\circ$ so that $(G, \circ, \star)$ is a skew brace with additive group $(G, \star)$ and also with additive group $(G, \circ)$. If $G$ is a skew brace, then $G$ corresponds ... More

Classification of simple weight modules for the $N=2$ superconformal algebraApr 18 2019In this paper, we classify all simple weight modules with finite dimensional weight spaces over the $N=2$ superconformal algebra. As an application, we give a new proof of the classification of such modules for the $N=1$ superconformal algebra, which ... More

A Hopf Algebra from Preprojective ModulesApr 17 2019Let $Q$ be a finite type quiver i.e. ADE Dynkin quiver. Denote by $\Lambda$ its preprojective algebra. It is known that there are finitely many indecomposable $\Lambda$-modules if and only if $Q$ is of type $A_1,A_2,A_3,A_4$. In this paper, extending ... More

Type $\tilde{C}$ Temperley-Lieb algebra quotients and Catalan combinatoricsApr 17 2019We study some algebraic and combinatorial features of two algebras that arise as quotients of Temperley-Lieb algebras of type $\tilde{C}$, namely, the two-boundary Temperley-Lieb algebra and the symplectic blob algebra. We provide a monomial basis for ... More

A class of nilpotent Lie algebras whose center acts nontrivially in cohomologyApr 17 2019We show that the central representation is nontrivial for all one-dimensional central extensions of nilpotent Lie algebras possessing a codimension one abelian ideal.