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preprint2026arXiv

Half of finite abelian groups are unit groups

A group is called realizable if it is the group of units in a ring with identity. The classification of realizable groups is a difficult open problem -- originally posed by László Fuchs -- and is an active area of research. Realizable groups seem rare, but their proportion within a fixed class of groups (cyclic, dihedral, finite abelian, etc.) varies. To quantify this proportion, we introduce the realizable density of a class of finite groups as an analog of natural density for subsets of the natural numbers. The realizable finite cyclic groups and the realizable finite abelian $p$ -groups for $p$ odd have been classified; we prove that their realizable densities are 1/4 and 0, respectively. The realizable finite abelian 2-groups -- and more generally the realizable finite abelian groups -- have not been fully classified, and these special cases appear quite difficult. Nonetheless, we prove that the realizable density of finite abelian 2-groups is 1 and the realizable density of finite abelian groups is 1/2. Our work combines existing classification theorems for realizable groups with tools from analytic number theory.

preprint2017arXiv

On Sweedler's cofree cocommutative coalgebra

We give a direct proof of a result of Sweedler describing the cofree cocommutative coalgebra over a vector space, and use our approach to give an explicit construction of liftings of maps into this universal coalgebra. The basic ingredients in our approach are local cohomology and residues.

preprint2017arXiv

Subregular subalgebras and invariant generalized complex structures on Lie groups

We introduce the notion of a subregular subalgebra, which we believe is useful for classification of subalgebras of Lie algebras. We use it to construct a non-regular invariant generalized complex structure on a Lie group. As an illustration of the study of invariant generalized complex structures, we compute them all for the real forms of G2.

preprint2012arXiv

Atomistic subsemirings of the lattice of subspaces of an algebra

Let A be an associative algebra with identity over a field k. An atomistic subsemiring R of the lattice of subspaces of A, endowed with the natural product, is a subsemiring which is a closed atomistic sublattice. When R has no zero divisors, the set of atoms of R is endowed with a multivalued product. We introduce an equivalence relation on the set of atoms such that the quotient set with the induced product is a monoid, called the condensation monoid. Under suitable hypotheses on R, we show that this monoid is a group and the class of k1_A is the set of atoms of a subalgebra of A called the focal subalgebra. This construction can be iterated to obtain higher condensation groups and focal subalgebras. We apply these results to G-algebras for G a group; in particular, we use them to define new invariants for finite-dimensional irreducible projective representations.

preprint2017arXiv

Quasi-duo differential polynomial rings

In this article we give a characterization of left (right) quasi-duo differential polynomial rings. In particular, we show that a differential polynomial ring is left quasi-duo if and only if it is right quasi-duo. This yields a partial answer to a question posed by Lam and Dugas in 2005. We provide non-trivial examples of such rings and give a complete description of the maximal ideals of an arbitrary quasi-duo differential polynomial ring. Moreover, we show that there is no left (right) quasi-duo differential polynomial ring in several indeterminates.

preprint2017arXiv

Computing explicit isomorphisms with full matrix algebras over $\mathbb{F}_q(x)$

We propose a polynomial time $f$ -algorithm (a deterministic algorithm which uses an oracle for factoring univariate polynomials over $\mathbb{F}_q$ ) for computing an isomorphism (if there is any) of a finite dimensional $\mathbb{F}_q(x)$ -algebra $A$ given by structure constants with the algebra of $n$ by $n$ matrices with entries from $\mathbb{F}_q(x)$ . The method is based on computing a finite $\mathbb{F}_q$ -subalgebra of $A$ which is the intersection of a maximal $\mathbb{F}_q[x]$ -order and a maximal $R$ -order, where $R$ is the subring of $\mathbb{F}_q(x)$ consisting of fractions of polynomials with denominator having degree not less than that of the numerator.

preprint2016arXiv

Approximation of Gram-Schmidt Orthogonalization by Data Matrix

For a matrix ${\bf A}$ with linearly independent columns, this work studies to use its normalization $\bar{\bf A}$ and ${\bf A}$ itself to approximate its orthonormalization $\bf V$ . We theoretically analyze the order of the approximation errors as $\bf A$ and $\bar{\bf A}$ approach ${\bf V}$ , respectively. Our conclusion is able to explain the fact that a high dimensional Gaussian matrix can well approximate the corresponding truncated Haar matrix. For applications, this work can serve as a foundation of a wide variety of problems in signal processing such as compressed subspace clustering.

preprint2016arXiv

Core and Dual Core Inverses of a Sum of Morphisms

Let $\mathscr{C}$ be an additive category with an involution $\ast$ . Suppose that $φ: X \rightarrow X$ is a morphism of $\mathscr{C}$ with core inverse $φ^{\co} : X \rightarrow X$ and $η: X \rightarrow X$ is a morphism of $\mathscr{C}$ such that $1_X+φ^{\co}η$ is invertible. Let $α=(1_X+φ^{\co}η)^{-1},$ $β=(1_X+ηφ^{\co})^{-1},$ $\varepsilon=(1_X-φφ^{\co})ηα(1_X-φ^{\co}φ),$ $γ=α(1_X-φ^{\co}φ)β^{-1}φφ^{\co}β,$ $σ=αφ^{\co}φα^{-1}(1_X-φφ^{\co})β,$ $δ=β^{\ast}(φ^{\co})^{\ast}η^{\ast}(1_X-φφ^{\co})β.$ Then $f=φ+η-\varepsilon$ has a core inverse if and only if $1_X-γ$ , $1_X-σ$ and $1_X-δ$ are invertible. Moreover, the expression of the core inverse of $f$ is presented. Let $R$ be a unital $\ast$ -ring and $J(R)$ its Jacobson radical, if $a\in R^{\co}$ with core inverse $a^{\co}$ and $j\in J(R)$ , then $a+j\in R^{\co}$ if and only if $(1-aa^{\co})j(1+a^{\co}j)^{-1}(1-a^{\co}a)=0$. We also give the similar results for the dual core inverse.

preprint2017arXiv

Enveloping algebras that are principal ideal rings

Let $L$ be a restricted Lie algebra over a field of positive characteristic. We prove that the restricted enveloping algebra of $L$ is a principal ideal ring if and only if $L$ is an extension of a finite-dimensional torus by a cyclic restricted Lie algebra.

preprint2015arXiv

Algebraic Conditions for Generating Accurate Adjacency Arrays

Data processing systems impose multiple views on data as it is processed by the system. These views include spreadsheets, databases, matrices, and graphs. Associative arrays unify and simplify these different approaches into a common two-dimensional view of data. Graph construction, a fundamental operation in the data processing pipeline, is typically done by multiplying the incidence array representations of a graph, $\mathbf{E}_\mathrm{in}$ and $\mathbf{E}_\mathrm{out}$ , to produce an adjacency matrix of the graph that can be processed with a variety of machine learning clustering techniques. This work focuses on establishing the mathematical criteria to ensure that the matrix product $\mathbf{E}_\mathrm{out}^\intercal\mathbf{E}_\mathrm{in}$ is the adjacency array of the graph. It will then be shown that these criteria are also necessary and sufficient for the remaining nonzero product of incidence arrays, $\mathbf{E}_\mathrm{in}^\intercal\mathbf{E}_\mathrm{out}$ to be the adjacency matrices of the reversed graph. Algebraic structures that comply with the criteria will be identified and discussed.

preprint2017arXiv

2-D Skew Constacyclic Codes Over $R[x,y,ρ,θ]$

In this paper, we give some characterizations of the ring $R[x,y,ρ,θ]$ where R is a commutative ring. We investigate 2-D skew $(λ_1,λ_2)$ -constacyclic codes.

preprint2016arXiv

The derived Picard group of an affine Azumaya algebra

We describe the derived Picard group of an Azumaya algebra A on an affine scheme X in terms of global sections of the constant sheaf of integers on X, the Picard group of X, and the stabilizer of the Brauer class of A under the action of Aut(X). In particular, we find that the derived Picard group of an Azumaya algebra is generally not isomorphic to that of the underlying scheme. In the case of the trivial Azumaya algebra, our result refines Yekutieli's description of the derived Picard group of a commutative algebra. We also get, as a corollary, an alternate proof of a result of Antieau which relates derived equivalences to Brauer equivalences for affine Azumaya algebras. The example of a Weyl algebra in finite characteristic is examined in some detail.

preprint2017arXiv

Prime and primitive Kumjian-Pask algebras

In this paper, prime as well as primitive Kumjian-Pask algebras $\mathrm{KP}_R(Λ)$ of a row-finite $k$ -graph $Λ$ over a unital commutative ring $R$ are completely characterized in graph-theoretic and algebraic terms. By applying quotient $k$ -graphs, these results describe prime and primitive graded basic ideals of Kumjian-Pask algebras. In particular, when $Λ$ is strongly aperiodic and $R$ is a field, all prime and primitive ideals of a Kumjian-Pask algebra $\mathrm{KP}_R(Λ)$ are determined.

preprint2017arXiv

Fukaya categories in Koszul duality theory

In this paper, we define $A_{\infty}$ -Koszul duals for directed $A_{\infty}$ -categories in terms of twists in their $A_{\infty}$ -derived categories. Then, we compute a concrete formula of $A_{\infty}$ -Koszul duals for path algebras with directed $A_n$ -type Gabriel quivers. To compute an $A_\infty$ -Koszul dual of such an algebra $A$ , we construct a directed subcategory of a Fukaya category which are $A_\infty$ -derived equivalent to the category of $A$ -modules and compute Dehn twists as twists. The formula unveils all the ext groups of simple modules of the parh algebras and their higher composition structures.

preprint2013arXiv

Monoidal ring and coring structures obtained from wreaths and cowreaths

Let $A$ be an algebra in a monoidal category $\Cc$, and let $X$ be an object in $\Cc$. We study $A$ -(co)ring structures on the left $A$ -module $A\ot X$. These correspond to (co)algebra structures in $EM(\Cc)(A)$, the Eilenberg-Moore category associated to $\Cc$ and $A$ . The ring structures are in bijective correspondence to wreaths in $\Cc$, and their category of representations is the category of representations over the induced wreath product. The coring structures are in bijective correspondence to cowreaths in $\Cc$, and their category of corepresentations is the category of generalized entwined modules. We present several examples coming from (co)actions of Hopf algebras and their generalizations. Various notions of smash products that have appeared in the literature appear as special cases of our construction.

preprint2016arXiv

Central reflections and nilpotency in exact Mal'tsev categories

We study nilpotency in the context of exact Mal'tsev categories taking central extensions as the primitive notion. This yields a nilpotency tower which is analysed from the perspective of Goodwillie's functor calculus. We show in particular that the reflection into the subcategory of $n$ -nilpotent objects is the universal endofunctor of degree $n$ if and only if every $n$ -nilpotent object is $n$ -folded. In the special context of a semi-abelian category, an object is $n$ -folded precisely when its Higgins commutator of length $n+1$ vanishes.

preprint2017arXiv

Purely infinite simple Kumjian-Pask algebras

Given any finitely aligned higher-rank graph $Λ$ and any unital commutative ring $R$ , the Kumjian-Pask algebra $\mathrm{KP}_R(Λ)$ is known as the higher-rank generalization of Leavitt path algebras. After characterizing simple Kumjian-Pask algebras by L.O. Clark and Y.E.P. Pangalela (and others), we focus in this article on the purely infinite simple ones. Briefly, we show that if $\mathrm{KP}_R(Λ)$ is simple and every vertex of $Λ$ is reached from a generalized cycle with an entrance, then $\mathrm{KP}_R(Λ)$ is purely infinite. We next prove a standard dichotomy for simple Kumjian-Pask algebras: in the case that each vertex of $Λ$ is reached only from finitely many vertices and $\mathrm{KP}_R(Λ)$ is simple, then $\mathrm{KP}_R(Λ)$ is either purely infinite or locally matritial. This result covers all unital simple Kumjian-Pask algebras.

preprint2014arXiv

A "q-deformed" generalization of the Hosszu-Gluskin theorem

In this paper a new form of the Hosszú-Gluskin theorem is presented in terms of polyadic powers and using the language of diagrams. It is shown that the Hosszú-Gluskin chain formula is not unique and can be generalized ("deformed") using a parameter q which takes special integer values. A version of the "q-deformed" analog of the Hosszú-Gluskin theorem in the form of an invariance is formulated, and some examples are considered. The "q-deformed" homomorphism theorem is also given.

preprint2016arXiv

Lie algebra configuration pairing

We give an algebraic construction of the topological graph-tree configuration pairing of Sinha and Walter beginning with the classical presentation of Lie coalgebras via coefficients of words in the associative Lie polynomial. Our work moves from associative algebras to preLie algebras to graph complexes, justifying the use of graph generators for Lie coalgebras by iteratively expanding the set of generators until the set of relations collapses to two simple local expressions. Our focus is on new computational methods allowed by this framework and the efficiency of the graph presentation in proofs and calculus involving free Lie algebras and coalgebras. This outlines a new way of understanding and calculating with Lie algebras arising from the graph presentation of Lie coalgebras.

preprint2016arXiv

Derived brackets for fat Leibniz algebras

Given a Leibniz algebra L with left center Z, we work on C(L,Z,S(Z)), the Z-standard complex of L with coefficients in S(Z). We construct the derived bracket for a fat Leibniz algebra in terms of a certain 3-cocycle and a Poisson algebra structure on the space of so-called "representable cochains".

preprint2016arXiv

$\mathbb{F}_p$ and $Z_p$ Valued Holomorphic Functions over Graphs

The definition of a holomorphic function over a general measurable space $S$ endowed with a Markov process is defined by Zeghib and Barre. In this article we consider holomorphic functions over graphs whose ranges are a given finite field or a cyclic group. Also we consider a relation between $\mathbb{C}$ -holomorphic functions over regular trees and the field of $p$ -adic numbers.

preprint2010arXiv

Extensions of witness mappings

We deal with the problem of coexistence in interval effect algebras using the notion of a witness mapping. Suppose that we are given an interval effect algebra $E$ , a coexistent subset $S$ of $E$ , a witness mapping $β$ for $S$ , and an element $t\in E\setminus S$ . We study the question whether there is a witness mapping $β_t$ for $S\cup\{t\}$ such that $β_t$ is an extension of $β$ . In the main result, we prove that such an extension exists if and only if there is a mapping $e_t$ from finite subsets of $S$ to $E$ satisfying certain conditions. The main result is then applied several times to prove claims of the type "If $t$ has a such-and-such relationship to $S$ and $β$ , then $β_t$ exists".

preprint2016arXiv

Weak clean index of a ring

Motivated by the concept of clean index of rings, we introduce the concept of weak clean index of rings. For any element $a$ of a ring $R$ with unity, we define $ χ(a)=\{e\in R\mid e^2=e\text{ and }a-e \mbox{ or } a+e \mbox{ is a unit}\}$. The weak clean index of $R$ is defined as $\sup \{|χ(a)|: a\in R\}$ and it is denoted by $\win(R),$ where $| χ(a)|$ denotes the cardinality of the set $χ(a)$ . In this article, we characterize rings of weak clean indices $1$ , $2$ and $3$ .

preprint2016arXiv

A Dichotomy for GK dimensions of simple modules over simple differential rings

The Gelfand--Kirillov dimension has gained importance since its introduction as an tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand--Kirillov dimension of simple modules over certain simple rings of differential operators.

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