text large_stringlengths 7 3.02k |
|---|
This article is part of my upcoming masters thesis which investigates the following open problem from the book, Free Lattices, by <a href="http://R.Freese" rel="external noopener nofollow" class="link-external link-http">this http URL</a>, <a href="http://J.Jezek" rel="external noopener nofollow" class="link-external l... |
Let $D$ be a noncommutative division ring. In a recent paper, Lee and Lin proved that if $\text{char}\, D\ne 2$, the only solution of additive maps $f, g$ on $D$ satisfying the identity $f(x) = x^n g(x^{-1})$ on $D\setminus \{0\}$ with $n\ne 2$ a positive integer is the trivial case, that is, $f=0$ and $g=0$. |
In this paper, first we introduce the notions of 3-tri-Leibniz algebras and embedding tensors on 3-Leibniz algebras. We show that an embedding tensor gives rise to a 3-tri-Leibniz algebra. |
Commutative hypercomplex algebras offer significant advantages over traditional quaternions due to their compatibility with linear algebra techniques and efficient computational implementation, which is crucial for broad applicability. This paper explores a novel family of commutative hypercomplex algebras, referred t... |
For a given abelian group G, we classify the isomorphism classes of G-gradings on the simple Lie algebras of types A_n (n >= 1), B_n (n >= 2), C_n (n >= 3) and D_n (n > 4), in terms of numerical and group-theoretical invariants. The ground field is assumed to be algebraically closed of characteristic diffe... |
We examine when division algebras can share common splitting fields of certain types. In particular, we show that one can find fields for which one has infinitely many Brauer classes of the same index and period at least 3, all nonisomorphic and having the same set of finite splitting fields as well as the same splitt... |
A semigroup is called $E$-$separated$ if for any distinct idempotents $x,y\in X$ there exists a homomorphism $h:X\to Y$ to a semilattice $Y$ such that $h (x)\ne h |
Let $\mathcal O$ be the ring of integers in a finite extension of $\mathbb Q_p$. If $G$ is a finite group and $\Gamma$ is a maximal order containing the group ring $\mathcal O[G]$, Jacobinski's conductor formula gives a complete description of the central conductor of $\Gamma$ into $\mathcal O[G]$ in terms of char... |
We consider the translational hull $\Omega(I)$ of an arbitrary subsemigroup $I$ of an endomorphism monoid $\mathrm{End}(A)$ where $A$ is a universal algebra. We give conditions for every bi-translation of $I$ to be realised by transformations, or by endomorphisms, of $A$. |
We find explicitly the generating functions of the multiplicities in the pure and mixed trace cocharacter sequences of two $4\times 4$ matrices over a field of characteristic 0. We determine the asymptotic behavior of the multiplicities and show that they behave as polynomials of 14th degree. |
We continue our study of topological partial *-algebras, focusing our attention to *-semisimple partial *-algebras, that is, those that possess a {multiplication core} and sufficiently many *-representations. We discuss the respective roles of invariant positive sesquilinear (ips) forms and representable continuous li... |
Hochschild cohomology is crucial for understanding deformation theory. In <a href="https://arxiv.org/abs/2304.10223" data-arxiv-id="2304.10223" class="link-https">arXiv:2304.10223</a>, we have computed the Hochschild cohomology for gentle algebras of punctured surfaces. |
Let $K$ be a complete discrete valued field of characteristic $p$ with residue $k$ which is not necessarily perfect. We prove the Conjecture in \cite{cs} that a $p$-algebra over $K$ contains a totally ramified cyclic maximal subfield if it contains a totally ramified purely inseparable maximal subfield provided $k$ sa... |
The aim of the papers is to describe the left regular left quotient ring ${}'Q(R)$ and the right regular right quotient ring $Q'(R)$ for the following algebras $R$: $\mS_n=\mS_1^{\t n}$ is the algebra of one-sided inverses, where $\mS_1=K\langle x,y\, | \, yx=1\rangle$, $\CI_n=K\langle \der_1, \ldots, \der_n,\i... |
In this paper, we continue the study of weak multiplier Hopf algebras. We recall the notions of the source and target maps $\varepsilon_s$ and $\varepsilon_t$, as well as of the source and target algebras. |
The affine space of traceless complex matrices in which the sum of all elements in every row and every column is equal to one is presented as an example of an affine space with a Lie bracket or a Lie affgebra. |
Let $b$ be a non-degenerate symmetric (respectively, alternating) bilinear form on a finite-dimensional vector space $V$, over a field with characteristic different from $2$. In a previous work, we have determined the maximal possible dimension for a linear subspace of $b$-alternating (respectively, $b$-symmetric) nil... |
In this paper we generalize the classical Groebner basis technique to prove the existence and present a method of computation of a dimension polynomial in two variables associated with a finitely generated D-module, that is, a finitely generated module over a Weyl algebra. We also present corresponding algorithms and ... |
We classify the category of finite-dimensional real division composition algebras having a non-abelian Lie algebra of derivations. Our complete and explicit classification is largely achieved by introducing the concept of a quasi-description of a category, and using it to express the problem in terms of normal form pr... |
Let $K$ be an algebraically closed field of characteristic zero, and let $A$ and $B$ be two simple algebras with involution over $K$. In this note we study the embedding problem for algebras with involution. |
This article is devoted to rational equivalence for non-commutative polynomial algebras in a context including both the classical Gelfand-Kirillov problem and its quantum version. We introduce in this ``mixed'' context some reference algebras and define two new invariants allowing us to separate the fraction f... |
We classify group gradings on the simple Lie algebras of types $G_2$ and $D_4$ over the field of real numbers (or any real closed field): fine gradings up to equivalence and $G$-gradings, with a fixed group $G$, up to isomorphism. |
Let K,S,D be a division ring, an endomorphism and a S-derivation of K, respectively. In this setting we introduce generalized noncommutative symmetric functions and obtain Vieta formula and decompositions of differential operators. |
It is known that a lattice is representable as a ring of sets iff the lattice is distributive. CRL is the class of bounded distributive lattices (DLs) which have representations preserving arbitrary joins and meets. |
Every quasi completely regular semiring is a b-lattice of completely Archimedean semirings, i.e., a b-lattice of nil-extensions of completely simple semirings. In this paper we consider the semiring which is a b-lattice of nil-extensions of orthodox completely simple semirings. |
Let $\Lambda$ be an artin algebra and $\mathfrak{A}$ a two-sided idempotent ideal of $\Lambda$, that is, $\mathfrak{A}$ is the trace of a projective $\Lambda$-module $P$ in $\Lambda$. We consider the categories of finitely generated modules over the associated rings $\Lambda/\mathfrak{A}, \Lambda$ and $\Gamma=\mathrm{... |
We establish Gröbner--Shirshov bases theory for commutative dialgebras. We show that for any ideal $I$ of $Di[X]$, $I$ has a unique reduced Gröbner--Shirshov basis, where $Di[X]$ is the free commutative dialgebra generated by a set $X$, in particular, $I$ has a finite Gröbner--Shirshov basis if $X$ is finite. As applic... |
In \cite{F}, Faith asked for what rings $R$ does the Dual Baer Criterion hold in Mod-$R$, that is, when does $R$-projectivity imply projectivity for all right $R$-modules? Such rings $R$ were called right testing. |
In this paper, we establish the Composition-Diamond lemma for $\lambda$-differential associative algebras over a field $K $ with multiple operators. As applications, we obtain Gröbner-Shirshov bases of free $\lambda$-differential Rota-Baxter algebras. |
In max algebra it is well-known that the sequence A^k, with A an irreducible square matrix, becomes periodic at sufficiently large k. This raises a number of questions on the periodic regime of A^k and A^k x, for a given vector x. |
Let S be a symbol algebra. The trace form of S is computed and it is shown how this form can be used to determine whether S is a division algebra or not. |
Motivated by the orthogonality relations for irreducible characters of a finite group, we evaluate the sum of a finite group of linear characters of a Hopf algebra, at all grouplike and skew-primitive elements. We then discuss results for products of skew-primitive elements. |
Many important quantum algebras such as quantum symplectic space, quantum Euclidean space, quantum matrices, $q$-analogs of the Heisenberg algebra and the quantum Weyl algebra are semi-commutative. In addition, enveloping algebras $U(L_+)$ of even Lie color algebras are also semi-commutative. |
Let (S,L) be a Lie-Rinehart algebra over a commutative ring R. This article proves that, if S is flat as an R-module and has Van den Bergh duality in dimension n, and if L is finitely generated and projective with constant rank d as an S-module, then the enveloping algebra of (S,L) has Van den Bergh duality in dimensio... |
It is known that any symmetric matrix $M$ with entries in $\R[x]$ and which is positive semi-definite for any substitution of $x\in\R$, has a Smith normal form whose diagonal coefficients are constant sign polynomials in $\R[x]$. We generalize this result by considering a symmetric matrix $M$ with entries in a formall... |
For finite-dimensional Hopf algebras, their classification in characteristic $0$ (e.g. over $\mathbb{C}$) has been investigated for decades with many fruitful results, but their structures in positive characteristic have remained elusive. In this paper, working over an algebraically closed field $\mathbf{k}$ of prime ... |
The paper is dedicated to the study of certain non commutative graded AS Gorenstein algebras $\Lambda $. <br>The main result of the paper is that for Koszul algebras $\Lambda $ with Yoneda algebra $\Gamma $, such that both $\Lambda $ and $\Gamma $ are graded AS Gorenstein noetherian of finite local cohomology dimensio... |
In this paper, we give a natural braiding on the universal central extension of a crossed module of Lie algebras with a given braiding and construct the universal central extension of a braided crossed module of Lie algebras, showing that, when one of the constructions exists, both exist and coincide. |
We consider projective, irreducible, non-singular curves over an algebraically closed field $\k$. A cover $Y \to X$ of such curves corresponds to an extension $\Omega/\Sigma$ of their function fields and yields an isomorphism $\A_{Y} \simeq \A_{X} \otimes_{\Sigma} \Omega$ of their geometric adele rings. |
As an alternative perspective on the injectivity of a pure-injective module, a pure-injective module M is said to be pi-indigent if its subinjectivity domain is smallest possible, namely, consisting of exactly the absolutely pure modules. A module M is called subinjective relative to a module N if for every extension ... |
This is an old paper put here for archeological purposes. It is proved that a finite-dimensional Lie algebra over a field of characteristic p>5, that can be written as a vector space (not necessarily direct) sum of two nilpotent subalgebras, is solvable. |
In this paper we introduce a new invariant for a non-degenerate evolution algebra, which consists of an ordered sequence of evolution algebras of lower dimension, belonging all of them to a specific family. We use this invariant to propose a method to classify non-degenerate evolution algebras, and we apply it up to d... |
Consider a finite-dimensional, complex Lie algebra G and a semi-simple automorphism {\alpha}. This note aims to give a short and simple proof for explicit upper bounds for the derived length of the radical R and the rank of a Levi complement G/R in terms of the number of eigenvalues of {\alpha} and the dimension of th... |
We extend the classical associative PI-theory to Associative Pairs, and in doing so, we introduce related notions already present for algebras (and Jordan systems) as the ones of PI-element and PI-ideal, extended centroid and central closure. |
It is proved that the ring $R$ with center $Z(R)$, such that the module $R_{Z(R)}$ is an essential extension of the module $Z(R)_{Z(R)}$, is not necessarily right quasi-invariant, i.e., maximal right ideals of the ring $R$ are not necessarily ideals. We use central essentiality to obtain conditions which are sufficien... |
We define when a noetherian ring R is called a right ( or a left) weakly krull symmetric ring . We then prove that if R is a right ( or a left ) krull homogenous ring then R is a right ( or a left ) weakly krull symmetric ring . |
We present a geometrically oriented classification theory for non-Abelian extensions of groupoids generalizing the classification theory for Abelian extensions of groupoids by Westman as well as the familiar classification theory for non-Abelian extensions of groups by Schreier and Eilenberg-MacLane. As an application... |
We present an algebraic theory of orthogonal polynomials in several variables that includes classical orthogonal polynomials as a special case. Our bottom line is a straightforward connection between apolarity of binary forms and the inner product provided by a linear functional defined on a polynomial ring. |
In this paper we study algebras acted on by a finite group $G$ and the corresponding $G$-identities. Let $M_2( \mathbb{C})$ be the $2\times 2$ matrix algebra over the field of complex numbers $ \mathbb{C}$ and let $sl_2( \mathbb{C})$ be the Lie algebra of traceless matrices in $M_2( \mathbb{C})$. |
We consider $14$ medial-like balanced functional equations with four object variables for a pair $(f, g)$ of binary quasigroup operations. Then, we prove that every algebra $(B; f, g)$ with quasigroup operations satisfying a medial-like balanced functional equation has a linear representation on an abelian group $(B; ... |
We consider a class of quasi-Hopf algebras which we call \emph{generalized twisted quantum doubles}. They are abelian extensions $H = \mb{C}[\bar{G}] \bowtie \mb{C}[G]$ ($G$ is a finite group and $\bar{G}$ a homomorphic image), possibly twisted by a 3-cocycle, and are a natural generalization of the twisted quantum do... |
We consider algebras that can be realized as PBW deformations of (Artin-Schelter) regular algebras. This is equivalent to the homogenization of the algebra being regular. |
It is well known that an equivalence relation is invariant under the basic operations of an algebra if and only if it is invariant under the unary polynomials of the algebra. We show that a higher arity version of this property holds for a higher dimensional analogue of an equivalence relation. |
The algebra of GL_n-invariants of d-tuple of n x n nilpotent matrices with respect to the action by simultaneous conjugation is generated by the traces of products of nilpotent generic matrices in the case of an algebraically closed field of characteristic zero. We described a minimal separating set for this algebra i... |
This introduction to graphs and graph algebras provides the optimal bound for the number of all paths of length $k$ in a graph with $N\geq k$ edges and no loops. Our proof relies on a construction of a number of terminating algorithms that reshape such graphs without ever decreasing the number of paths of length $k$. |
From a system consisting of a right non-degenerate ring $R$, a pair of $R$-bimodules $Q$ and $P$ and an $R$-bimodule homomorphism $\psi:P\otimes Q\longrightarrow R$ we construct a $\Z$-graded ring $\mathcal{T}_{(P,Q,\psi)}$ called the Toeplitz ring and (for certain systems) a $\Z$-graded quotient $\mathcal{O}_{(P,Q,\ps... |
Given a non-necessarily commutative unital ring $R$ and a unital partial representation $\Theta $ of a group $G$ into the Picard semigroup $\mathbf{PicS} (R)$ of the isomorphism classes of partially invertible $R$-bimodules, we construct an abelian group $\mathcal{C}(\Theta /R) $ formed by the isomorphism classes of pa... |
Let $U$ and $V$ be finite-dimensional vector spaces over an arbitrary field, and $\mathcal{S}$ be a subset of the space $\mathcal{L}(U,V)$ of all linear maps from $U$ to $V$. A map $F : \mathcal{S} \rightarrow V$ is called range-compatible when it satisfies $F(s) \in \mathrm{im}(s)$ for all $s \in \mathcal{S}$; it is ... |
$K$ is an algebraically closed field with characteristic $0$ and $f$ is a polynomial or a holomorphic function. We study all solutions of the equation $XA-AX=f(X)$, in the unknown $X\in M_n(K)$, when $A\in M_n(K)$ is diagonalizable. |
A congruence $\varepsilon$ on a semigroup $S$ is perfect if for any congruence classes $x\varepsilon$ and $y\varepsilon$ their product as subsets of $S$ coincides (as a set) with the congruence class $(xy)\varepsilon$. Perfect congruences on the bicyclic semigroup were found in \cite{key7}. |
A short proof is given to Dixmier's 6'th problem for the Weyl algebra (and other algebras of Gelfand-Kirillov dimension which is less than 3 like rings of differential operators on smooth irreducible algebraic curves). |
For complexes of modules we study two new constructions, which we call the pinched tensor product and the pinched Hom. They provide new methods for computing Tate homology and Tate cohomology, which lead to conceptual proofs of balancedness of Tate (co)homology for modules over associative rings. |
In this paper it is proved that the ideal $I_w$ of the weak polynomial identities of the superalgebra $M_{1,1}(E)$ is generated by the proper polynomials $[x_1,x_2,x_3]$ and $[x_2,x_1][x_3,x_1][x_4,x_1]$. This is proved for any infinite field $F$ of characteristic different from 2. |
The main aim of this paper is to study aggregation functions on lattices via clone theory approach. Observing that the aggregation functions on lattices just correspond to $0,1$-monotone clones, as the main result we show that for any finite $n$-element lattice $L$ there is a set of at most $2n+2$ aggregation function... |
In commutative invariant theory, a classical result due to Auslander says that if $R = \Bbbk[x_1, \dots, x_n]$ and $G$ is a finite subgroup of $\text{Aut}_{\text{gr}}(R) \cong \text{GL}(n,\Bbbk)$ which contains no reflections, then there is a natural graded isomorphism $R \hspace{1pt} \# \hspace{1pt} G \cong \text{End}... |
2-local derivation is a generalized derivation for a Lie algebra, which plays an important role to the study of local properties of the structure of the Lie algebra. In this paper, we prove that every 2-local derivation on the conformal Galilei algebra is a derivation. |
Let $K$ be an infinite field and $K< X> =K< X_1,...,X_n>$ the free associative algebra generated by $X=\{X_1,...,X_n\}$ over $K$. It is proved that if $I$ is a two-sided ideal of $K< X>$ such that the $K$-algebra $A=K< X> /I$ is almost commutative in the sense of [3], namely, with respect to it... |
This expository paper focuses on free Lie $K$-algebras and the basic PBW theorem. We argue in various ways that the basic PBW theorem is a quite close consequence of the Magnus-Witt theorems concerning free Lie algebras. |
We review some applications of Gröbner-Shirshov bases, including PBW theorems, linear bases of free universal algebras, normal forms for groups and semigroups, extensions of groups and algebras, embedding of algebras. |
Cosetal extensions of monoids generalise extensions of groups, special Schreier extensions of monoids and Leech's normal extensions of groups by monoids. They share a number of properties with group extensions, including a notion of Baer sum when the kernel is abelian. |
This paper aims to study the skew-symmetric super-biderivations of the special odd Hamiltonian superalgebra SHO(n, n; t). Let HO denote the odd Hamiltonian Lie superalgebra HO(n, n; t) and SHO the special odd Hamiltonian Lie superalgebra SHO(n, n; t) over a field of characteristic p > <a href="http://2.Utilizing" r... |
We consider graded twisted Calabi-Yau algebras of dimension 3 which are derivation-quotient algebras of the form $A = \kk Q/I$, where $Q$ is a quiver and $I$ is an ideal of relations coming from taking partial derivatives of a twisted superpotential on $Q$. We define the type $(M, P, d)$ of such an algebra $A$, where ... |
This article is a sequel of [4], where we introduced quadratic forms on a module~ $V$ over a supertropical semiring $R$ and analysed the set of bilinear companions of a quadratic form $q: V \to R$ in case that the module $V$ is free, with fairly complete results if $R$ is a supersemifield. Given such a companion $b$ w... |
In this paper we investigate classifications of all (transposed) Poisson algebras of the associated associative null-filiform algebra |
We develop a theory which unifies the universal (co)acting bi/Hopf algebras as studied by Sweedler, Manin and Tambara with the recently introduced \cite{AGV1} bi/Hopf-algebras that are universal among all support equivalent (co)acting bi/Hopf algebras. Our approach uses vector spaces endowed with a family of linear ma... |
The present article represents a step forward in the study of the following problem: If $\mathbb{A}=(A_{1},A_{2})$ and $\mathbb{H}=(H_{1},H_{2})$ are Hopf braces in a symmetric monoidal category C such that $(A_{1},H_{1})$ and $(A_{2},H_{2})$ are matched pairs of Hopf algebras, then we want to know under what condition... |
In this short note, we give a description of semisimple orbits in the restricted Cartan type Lie algebras $W, S, H$. |
Quotient grading classes are essential participants in the computation of the intrinsic fundamental group $\pi_1(A)$ of an algebra $A$. In order to study quotient gradings of a finite-dimensional semisimple complex algebra $A$ it is sufficient to understand the quotient gradings of twisted gradings. |
Let $A$ be a Koszul Artin-Schelter regular algebra and $B=A_P[y_1,y_2;\varsigma,\nu]$ be a graded double Ore extension of $A$ where $\varsigma:A\to M_{2\times 2}(A)$ is a graded algebra homomorphism and $\nu:A\to A^{\oplus 2}$ is a degree one $\varsigma$-derivation. We construct a minimal free resolution for the trivi... |
Starting from an operad, one can build a family of posets. From this family of posets, one can define an incidence Hopf algebra. |
We describe Rota-Baxter operators on split octonions. It turns out that up to some transformations there exists exactly one such non-splitting operator over any field. |
We will give several reduction theorems for Noether's problem. |
We describe how to construct all inverse semigroups Morita equivalent to a given inverse semigroup. This is done by taking the maximum inverse images of the regular Rees matrix semigroups over the inverse semigroup where the sandwich matrix satisfies what we call the McAlister conditions. |
The Lie algebra version of the Krull-Schmidt Theorem is formulated and proved. This leads to a method for constructing the automorphisms of a direct sum of Lie algebras from the automorphisms of its indecomposable components. |
We introduce the notion of differential torsors, which allows the adaptation of constructions from algebraic geometry to differential Galois theory. Using these differential torsors, we set up a general framework for applying patching techniques in differential Galois theory over fields of characteristic zero. |
We introduce a total order and the absolute value function for dual numbers. The absolute value function of dual numbers are with dual number values, and have properties similar to the properties of the absolute value function of real numbers. |
In this paper we count the number of k-potent elements over HZp , the quaternion algebra over Zp, and we give a descriptive formula for the general case. For k in {3, 4, 5}, we give an explicit formula for these values. |
Leavitt path algebras are shown to be algebras of right quotients of their corresponding path algebras. Using this fact we obtain maximal algebras of right quotients from those (Leavitt) path algebras whose associated graph satisfies that every vertex connects to a line point (equivalently, the Leavitt path algebra ha... |
This note is devoted to the construction of two very easy examples, of respective dimensions 4 and 6, of graded Lie algebras whose grading is not given by a semigroup, the latter one being a semisimple algebra. It is shown that 4 is the minimal possible dimension. |
In this paper, we investigate cohomologies and non-abelian extensions of lambda-weighted Rota-Baxter Lie coalgebras. First, we consider Lie comodules and cohomologies of lambda-weighted Rota-Baxter Lie coalgebras. |
In this paper, we study positive as well as non-negative and non-trivial gradings on finite dimensional Lie algebras. We give a different proof that the existence of such a grading on a Lie algebra is invariant under taking field extensions, a result very recently obtained by Y. Cornulier. |
We describe how some aspects of abstract localization on module categories have applications to the study of injective comodules over some special types of corings. We specialize the general results to the case of Doi-Koppinen modules, generalizing previous results in this setting. |
We initiate a study of the growth and matrix-valued Hilbert series of non-negatively graded twisted Calabi-Yau algebras that are homomorphic images of path algebras of weighted quivers, generalizing techniques previously used to investigate Artin-Schelter regular algebras and graded Calabi-Yau algebras. Several result... |
We study the multiplication operation of square matrices over lattices. If the underlying lattice is distributive, then matrices form a semigroup; we investigate idempotent and nilpotent elements and the maximal subgroups of this matrix semigroup. |
Every Lie algebra over a field $E$ gives rise to new Lie algebras over any subfield $F \subseteq E$ by restricting the scalar multiplication. This paper studies the structure of these underlying Lie algebra in relation to the structure of the original Lie algebra, in particular the question how much of the original Li... |
In this paper we study the problem of finding explicit expressions for inner products on the space of complex square matrices $\Mn$. We show that, given an inner product $\lip \cdot, \cdot \rip$ on $\Mn$, with some conditions, there exist positive matrices $A_j$ and $B_j \in \Mn$, for $j=1, 2\dots, m$ such that $$ \li... |
The relationship between Jordan and Lie coalgebras is established. We prove that from any Jordan coalgebra $\langle A, \Delta\rangle$, it is possible to construct a Lie coalgebra $\langle L(A), \Delta_{L}\rangle$. |
In this thesis, we study the properties of a nonassociative algebra construction from skew polynomial rings. This construction was introduced by Petit in the 1960s but largely ignored until recently. |
A very first step to develop non-commutative algebraic geometry is the arithmetic of polynomials in non-commuting variables over a commutative field, that is, the study of elements in free associative algebras. This investigation is presented as a natural extension of the classical theory in one variable by using Leav... |
Let K be an arbitrary (commutative) field and L be an algebraic closure of it. Let V be a linear subspace of M_n(K), with n>2. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.