Papers

In this paper, we explore the idea that the graded Grothendieck group $K_0^{gr}$, or equivalently its positive cone, the talented monoid, can detect the structural type of higher-rank graph algebras (i.e., higher-rank graph $C^*$-algebras and Kumjian–Pask algebras). We show that the talented monoid captures some of the essential geometric information of a higher-rank graph, including the existence of cycles with and without entrances. In turn, we show that the graded $K$-theory can effectively distinguish the class of locally finite Kumjian–Pask algebras, and also the class of crossed product Kumjian–Pask algebras. We also derive talented monoid criteria for higher-rank graph algebras to be purely infinite simple, and not to be $AF$ or ultramatricial.

A recent conjecture states that a finite-dimensional complex evolution algebra is solvable if and only if it has no non-zero idempotents. We exhibit a three-dimensional counterexample over $\mathbb{C}$ whose isomorphism class already appears in the classification of three-dimensional complex evolution algebras. Since the conjecture is known in dimensions one and two, this counterexample has the smallest possible dimension. The algebra is defined over every field of characteristic different from $2$ and is the exceptional member of a one-parameter family of pairwise non-isomorphic non-solvable evolution algebras whose idempotents are determined explicitly. For the exceptional parameter, the derived series stabilises at a non-zero two-dimensional subalgebra whose only idempotent is zero. Direct sums with zero algebras give complex counterexamples in every dimension at least three. We also prove the conjectured equivalence whenever the stable term of the derived series is an evolution algebra.

We focus on the finite basis problem for flat semirings of the form $S(W)$, where $W$ is an arbitrary set of nonempty words. We prove that $S(W)$ generates a Cross variety (and hence is finitely based) whenever every word in $W$ has length at most $3$, whereas it is nonfinitely based whenever there exists $k \geq 3$ such that $W$ is $x^{k+2}$-free but not $x^{k+1}$-free. In particular, if $W_k$ denotes the set of all words of length $k$, then $S(W_k)$ is finitely based if and only if $k \leq 3$. Moreover, $S(W)$ is nonfinitely based whenever $W$ is finite and not $x^4$-free. These results provide a partial answer to an open problem raised by Jackson et al.~(J Algebra 611: 211–245, 2022).

We present an algebraic and geometric condition for bivariate quaternionic polynomials of arbitrary bidegree to have a univariate linear left or right factor. We apply this quaternionic factorization theorem to the bidegree (1,1) case and recover a classical theorem of Clifford in elliptic geometry. By applying to the bidegree (2,2) case, we obtain decompositions into two circles of celestial surfaces, namely surfaces in the 3-dimensional sphere that contain two circles through a general point. This results in an alternative proof and refinement for a theorem by Skopenkov and Krasauskas from 2019, which states that a non-quartic celestial surface is Möbius equivalent to either the pointwise product of circles in the unit-quaternions, or an inverse stereographic projection of the pointwise sum of circles in Euclidean space. Our proposed method extends this decomposition result to the quartic case and we show that surfaces are, up to Möbius equivalence and stereographic projections, not both a sum and product of circles.

In the first part of this work, we have, for a not necessarily semisimple modular category, generalized the action of the mapping class groups of surfaces on the spaces of conformal blocks to the so-called derived block spaces. In the second part presented here, we compute this action explicitly in the case of Drinfel'd doubles of finite groups over fields of positive characteristic. To do that, we connect Lyubashenko's approach to mapping class group representations with the theory of representation varieties. In this way, we are able to show that the mapping class group representations on the derived block spaces are in general different from those on the ordinary block spaces.

Let $L$ be a Lie algebra acting by derivations on an associative algebra $A$ over a field $F$ of characteristic zero. The polynomial identities satisfied by $A$ with respect to this action are called differential identities, or $L$-identities. In this paper, we study the differential identities of the algebra $UT_k$ of $k\times k$ upper triangular matrices and take a first step toward the classification of minimal $L$-varieties of differential exponent $3$. We first prove that, whenever $UT_k$ generates a minimal variety of algebras with derivations, the $L$-action can be replaced by its semisimple part. More precisely, it is enough to consider inner derivations induced by diagonal elements. We then apply this reduction to $UT_3$ and explicitly determine the $T_L$-ideal of differential identities and the corresponding differential codimension sequence for every such action on $UT_3$. Finally, we show that every $L$-variety generated by $UT_k$, with $k\geq 3$, contains $UT_3$ endowed with one of these $L$-actions.

This paper introduces the concept of e-Armendariz rings, a generalization of Armendariz rings via idempotent element $e$. A ring $R$ is called right e-Armendariz if for any polynomials $f(x), g(x) \in R[x]$, the condition $f(x)g(x)=0$ implies $a_ib_je=0$ for all coefficients $a_i$ of $f(x)$ and $b_j$ of $g(x)$. We establish that this new class of rings is not left-right symmetric and provide several illustrative examples. The main result of this paper provides a characterization that a ring $R$ is right e-Armendariz if and only if the idempotent $e$ is left semicentral and the corner ring $eRe$ is an Armendariz ring.

We introduce mutation-preserving generalized cluster algebras, for which the generalized cluster mutation in each direction is independent of the seed in the mutation equivalence class. We classify all irreducible generalized cluster algebras with this property. Then, a Markov-type Diophantine equation $x^2+y^2+z^2+2yz=kxyz$ is studied, which has a structure of the mutation-preserving generalized cluster algebra. We prove that positive integer solutions exist if and only if $1\leq k\leq 5$ and determine all their orbits under the associated generalized cluster mutation groups. In particular, multiple orbits occur for each $k=1,3$, whereas the solutions form a single orbit for each $ k=2,4,5$. We then classify all generalized Markov Laurent mutation invariants. As an application, a conjecture proposed by Chen-Li is proved, showing that every Laurent mutation invariant of irreducible sign-equivalent cluster algebras is essentially a polynomial in the corresponding basic invariant.

For a torsion-free abelian group $G$, let \[ K_{\mathbb{Q}}(G)=\operatorname{Frac} \mathbb{Q}[G] \] be the fraction field of its rational group algebra. We prove that this field determines the group up to isomorphism: \[ K_{\mathbb{Q}}(G) \cong K_{\mathbb{Q}}(H) \quad \Longleftrightarrow \quad G \cong H. \] The main structural input is that, over every field $k$ of characteristic zero, the monomial defect group \[ \Delta_k(G)=K_k(G)^\times/(k^\times X^G) \] is free abelian. We give a self-contained proof. It first treats the one-variable Puiseux field $F(t^{\mathbb{Q}})$: factorization from $F(t^{1/n!})$ to $F(t^{1/(n+1)!})$ gives split inclusions because the substituted irreducibles are square-free in characteristic zero. A transfinite decomposition of the divisible hull $\mathbb{Q} \otimes G$ then proves the general case. Given a field isomorphism, we compare the two monomial subgroups inside the common multiplicative group. Their intersection produces isomorphic subgroups $M\leq G$ and $N\leq H$, while the quotients $G/M$ and $H/N$ embed in free defect groups and are therefore free. Relative transcendence degree shows that these two free quotients have the same rank, completing the reconstruction. As consequences, rational-function stabilization of group fields exactly records free stabilization of groups, and Rickard's bounded sequence group yields a field $F$ with $F\cong F(x,y)$ but $F\not\cong F(x)$.

We study natural topologies on spaces of subobjects of a fixed object in a suitable category, with the aim of determining when these spaces are spectral. A central step in our approach, which is also of independent interest, is the comparison between the categorical notion of a $\lambda$-generated object and the order-theoretic notion of a $\lambda$-compact element in a lattice of subobjects. We prove that these notions coincide under some natural and mild assumptions . In the finitary case, this allows us to describe the finitely generated subobjects purely in order-theoretic terms and to construct, inside each interval of a subobject lattice, a canonical algebraic core. We study this construction abstractly for complete lattices and characterize it by a universal property. We then introduce the categorical Zariski topology on spaces of subobjects, relate it to the Scott topology, and obtain spectrality criteria for the whole subobject space and for its algebraic core.

A tensor has border rank at most $r$ if it can be written as $T=\lim_{\varepsilon \rightarrow 0} T(\varepsilon)$ where $T(\varepsilon)$ has rank at most $r$ for all sufficiently small $\varepsilon$. It is known that the map $\varepsilon \mapsto T(\varepsilon)$ can be assumed to be a (tensor valued) polynomial in $\varepsilon$. The smallest possible degree of such a map is called the error degree of $T$. The error degree and the related notion of order of degeneration are the two key quantities that we study in this paper. One motivation comes from debordering: by polynomial interpolation on the map $\varepsilon \mapsto T(\varepsilon)$ we can upper bound the tensor rank of $T$. For order 3 tensors, exponential upper bounds on the error degree and degeneration order were given almost 40 years ago in (Lehmkuhl Lickteig, 1989) and were not improved ever since. In this paper we give bounds that apply to a wide class of tensors, exponentially improving on (Lehmkuhl Lickteig, 1989). Our results are most general for tensors with 3 slices (format $m \times n \times 3$). In this case, our main assumption is on the rank of the matrix slices. We also give bounds that apply to arbitrary rectangular formats ($m \times n \times p$). In this case, we need an additional 1-regularity assumption on one of the slices of the tensor (recall that a matrix is said to be 1-regular if its eigenspaces are 1-dimensional). Under these assumptions we show that the error degree is at most 1, which yields a nontrivial debordering result (tensor rank at most $2r$ for border rank $r$). The results in (Lehmkuhl Lickteig, 1989) rely on an upper bound on the degree of the variety of tensors of border rank at most $r$. We rely instead on more specific properties of this algebraic variety, and in particular on commutativity properties of certain matrices derived from the tensor slices.

We prove that all (possibly infinitely generated) semi-Gorenstein-projective modules over a virtually Gorenstein Artin algebra are Gorenstein projective. We also establish a new criterion for determining the projective (injective) dimensions of modules over virtually Gorenstein Artin algebras. This enables us to show the validities of the Auslander-Gorenstein Conjecture, the (Strong) Nakayama Conjecture and Tachikawa's First Conjecture for virtually Gorenstein Artin algebras. Finally, we prove that the Auslander-Reiten Conjecture holds for virtually Gorenstein Artin algebras of finite CM-type.

A. A. Suslin and V. I. Kopeiko proved that the elementary linear, symplectic, and orthogonal groups are normal in the general linear group, symplectic group, and orthogonal group respectively. They also proved relative versions of these normality results with respect to an ideal of a ring. A. Bak, R. Basu, and R. A. Rao proved that the linear, symplectic, and orthogonal transvection groups are normal in the respective automorphism groups. These results are generalizations of the normality results by Suslin and Kopeiko in the setup of modules. In this paper, we prove stronger versions of the normality results proved by Bak, Basu, and Rao, which say that the relative transvection groups are normal in the respective automorphism groups.

For an odd prime $p$ and integers $r,q\ge 1$, we study the Butson-type complex Hadamard matrices $H_{p;r,q}(a,b)=\chi(1+p^r ab)$ on $\mathbb{Z}/p^q\mathbb{Z}$, where $\chi$ is a faithful character of the principal-unit quotient. The defining phase is the normalized $p$-adic logarithm, linear modulo $p^q$ exactly when $q\le r$ and nonlinear when $q>r$. We prove that this threshold simultaneously governs monomial equivalence to a finite abelian character table, the cocycle property of the canonical cyclic $2$-cochain, and projective scalarity of the canonical Weyl commutator. We classify the full projective monomial automorphism group as a three-parameter fractional-linear group of order $p^{2q}\varphi(p^q)$, with inverse limit a $\Gamma_0(p^r)$-type congruence subgroup of $\mathrm{PGL}_2(\mathbb{Z}_p)$. For $q>r$, scalar commutators survive on exactly $q-r+1$ maximal valuation-polarized windows; after quotienting radicals, each carries the standard perfect pairing on two cyclic groups of order $p^r$ and the associated finite Heisenberg extension. We determine the exact normalizers of these windows and prove distinct polarizations are pairwise nonconjugate inside the projective monomial group. Finally, abelianization separates $H_{p;r,q}$ from the Fourier matrix $F_{p^q}$ for every $q>r$; low-order defect and fingerprint computations illustrate why this group-theoretic invariant is stronger on the family.

We provide a general ansatz for the paracontrolled approach introduced by Gubinelli, Imkeller, Perkowski for treating singular SPDEs. The ansatz proposed covers a large class of equations. It is described via decorated trees that encode its coefficients and its stochastic data. The main novelty is the recursive definition of the paracontrolled stochastic iterated integrals that involve a well-chosen combinatorial set of words. The paralinearisation is performed via a lift of iterated paraproducts to modelled distributions and the reconstruction theorem coming from Regularity Structures. We also provide a fixed point argument and the renormalised equation with this ansatz when one uses the preparation map formalism that encompasses the BPHZ renormsalition. The last main result is the convergence of the renormalised stochastic data via the BPHZ renormalisation via an adaptation of the spectral gap approach. One obtains in the end a general solution theory for parabolic singular SPDEs via the paracontrolled approach.

For a field $F$ of characteristic zero with a derivation $\delta$, we provide a necessary and sufficient condition for a system of differential equations \begin{equation*} \delta y_1=f_0, \quad\delta y_2=f_1,\quad\dots\quad,\delta y_n=f_{n-1}, \end{equation*} where $f_0\in F$, $f_1\in F[y_1]$, $\dots$, $f_{n-1}\in F[y_1,\dots,y_{n-1}]$, to have elementary first integrals.

We study minimal noncommutative rings, that is noncommutative rings whose proper subrings and homomorphic images are all commutative. These rings were introduced by Bell and Danchev in order to test commutativity theorems. They raised the problems of describing all such rings in the finite and infinite cases. In the finite case, we give a classification into three pairwise disjoint classes, the first two of which are completely characterised. For the third class, we give a finite procedure which can produce any of (and only) the required rings. We also translate the problem into commutative algebra, in terms of finite local rings with small socle and a kind of cancellation property. Finally, we show that if an infinite minimal noncommutative ring exists, then it is a division algebra with very strange properties, and a counterexample to several longstanding conjectures.

To each VOA of strong CFT type, we associate a non-associative algebra which generalizes Griess algebras of One-Zero (OZ) VOAs. We compute the fusion law of Ising vectors in this algebra using techniques from Miyamoto, leveraging the module theory of the corresponding Virasoro VOA. This new class of algebras encapsulates the Chayet-Garibaldi algebras, a family of algebras constructed from absolutely simple linear algebraic groups that includes the $3876$-dimensional algebra for $E_8$.

Let $S$ be a semigroup with zero. This paper studies the zero-divisor digraph $\overrightarrow{\Gamma}(S)$ and the extended zero-divisor digraph $\overrightarrow{\Gamma}_{\!E}$, and introduces the annihilator digraph $\overrightarrow{\mathrm{AG}}(S)$ via left and right annihilators. The diameter bound when every zero-divisor is nilpotent, and the sinks and the sources of $\overrightarrow{\Gamma}_{\!E}(S)$ being identical to those of $\overrightarrow{\Gamma}(S)$ is demonstrated. The conditions in which $\overrightarrow{\Gamma}(S)=\overrightarrow{\Gamma}_{\!E}(S)=\overrightarrow{\mathrm{AG}}(S)$ holds are established, and the connectedness, diameter, girth, and vertex degrees of $\overrightarrow{\mathrm{AG}}(S)$ are bounded when every zero-divisor is nilpotent or two-sided. The extended zero-divisor digraph is connected if and only if the zero-divisor digraph is connected, and it contains a directed cycle if and only if the zero-divisor digraph does. The knit degrees of $\overrightarrow{\Gamma}(S)$, $\overrightarrow{\Gamma}_{\!E}(S)$, and $\overrightarrow{\mathrm{AG}}(S)$ are computed. For a unital ring $R$, the equality $\overrightarrow{\Gamma}_{\!E}(R)=\overrightarrow{\Gamma}(R)$ is characterized by nilpotency indices and one-sided annihilator conditions; it holds for the full matrix ring $M_{n}(F)$ over a field $F$ if and only if $n=2$, and $\overrightarrow{\mathrm{AG}}(M_{n}(F))$ is connected and contains a directed cycle. Moreover, for an artinian noncommutative ring $R$, it was proved that $\overrightarrow{\Gamma}(R)$ is connected if and only if $\overrightarrow{\Gamma}_{\!E}(R)$ is connected if and only if every one-sided identity element of $R$ is a two-sided identity of $R$.

We investigate core quandles and idempotents in quandle rings of core quandles. We answer several questions on the rank of core quandles and nontrivial idempotents in quandle rings. We also present solutions to two questions raised in a recent paper about non-trivial idempotents in quandle rings $\mathbb{Z}[R_5]$ and $\mathbb{Z}[C_5]$, where $\mathbb{Z}$, $R_5$ and $C_5$ are the ring of integers, the dihedral quandle of order 5, and the commutative quandle of order 5, respectively. We then study units in extended quandle rings of a trivial quandle and the Joyce quandle, and nilpotent elements in extended quandles rings of a trivial quandle, where the ground ring is an integral domain. As a consequence, we show that the quandle ring and the extended quandle ring of a trivial quandle are not nil clean rings. We also explore prime rings and semi-prime rings among quandle rings. We introduce zero-divisor graphs of quandle rings and find an intriguing mirror symmetry among the in-degree and out-degree of the vertices. Moreover, we find a bivariate polynomial in the ring $(\mathbb{Z}_{2n+1}[Q])[X,Y]$ that determines the commutative quandle of order $2n+1$, where $2n+1$ is prime.

T-quaternion rings are a special class of commutative unital subrings of the quaternion rings over commutative rings. Permutation Polynomials (PP) are typically defined over finite fields, such as $\mathbb{F}_{q}$, where $q$ is a prime power. This article examines permutation polynomials over finite T-quaternion rings, focusing specifically on linear, quadratic, and cubic cases. The study begins with permutation polynomials over the rings $\mathbb{Z}_{n}$, before extending the discussion to T-quaternion rings.

An extension theory for nearrings along the lines of the Schreier extension for groups and the Everett extension for rings is given. The semidirect sum of two nearrings is a special case of this theory. Already there are many examples of purpose built nearring constructions in the theory of nearrings which fall under this semidirect sum construction.

In 1991 Rickard asked whether every derived equivalence of algebras over a common field is standard. We construct a series of examples of non-standard derived equivalences, and thus answer the question negatively. Moreover, we give a necessary and sufficient condition for the question to be true.

For $n>2$ and $k>1$, define the polynomial \[F_{n,k}(x) = \Phi_n(1 + x + \cdots + x^k),\] where $\Phi_n$ denotes the $n$-th cyclotomic polynomial. The cyclotomic conjecture proposed by Gimbert (1999) exactly describes the irreducibility of $F_{n,k}(x)$ over $\mathbb{Q}$ in terms of $n$ and $k$. Conde, Gimbert, González, Miller and Miret (2014) established that the cyclotomic conjecture, if true, would imply the non-existence of almost Moore digraphs - a well-known open question concerning the directed degree-diameter problem. In this article, we prove the cyclotomic conjecture and, as a consequence, show that there are no almost Moore digraphs with maximum out-degree $d$ and diameter $k$ for any $d>1$ and $k>2$.

An explicit construction of a free nearring as well as the free product (= coproduct) of two nearrings are given in the variety of all, not necessarily zero-symmetric, nearrings.

The binary-tree height recursion defines an algebra $\mathcal{H}$ on $\mathbb{N}\cup\{-\infty\}$, with join given by $\max$ and product \[ a\star b=\max\{a,b\}+1. \] We show that weighted evaluation of a labelled tree depends only on the greatest depth of each label. Single-tree profiles are exactly the vectors satisfying the binary Kraft inequality, while finite joins realize every vector in $\bigl(\mathbb{N}\cup\{-\infty\}\bigr)^n$; hence the $n$-variable term operations form the free algebra $\mathcal{H}^n$. We also classify $\mathcal{H}$'s compatible semilattice operation, subalgebras, endomorphisms, congruences, and finite quotients, and recover the dyadic-composition spectrum at the full-linear boundary.

We characterize the subsets of a finite distributive lattice that arise as the minimizer sets of modular functions. It is immediate that the minimizer set of any modular function is a decomposition-closed sublattice. Our main theorem establishes the converse: every decomposition-closed sublattice is the minimizer set of a modular function. Our proof is constructive. Using Birkhoff's representation, we decompose the problem along the intervals determined by an arbitrary maximal chain of the given sublattice. The key ingredient is a weight assignment on connected difference posets that yields local modular functions vanishing precisely at the endpoints of each interval. We also provide an example showing that this characterization fails for nondistributive lattices.

We study the Gorenstein homological dimension $Ghd_k G$ of groups $G$ which are of type $FP_{\infty}$ over a commutative ring $k$. Our main result shows that, over an arbitrary ring $R$, every finitely presented Gorenstein flat $R$-module is projectively coresolved Gorenstein flat. Consequently, for a group $G$ of type $FP_\infty$ over $k$ with sfli$k<\infty$, the three natural dimensions for $G$, namely Gorenstein homological, Gorenstein cohomological and projectively coresolved Gorenstein flat, all coincide. Building on this collapse, we establish a Gorenstein homological analogue of Fel$'$dman's theorem, a formula for iterated $m$-fold self-extensions of a group $N$ of type $FP_\infty$ over $\mathbb Z$, and a field-detection theorem, showing that the Gorenstein homological dimension of a group of type $FP_\infty$ over a principal ideal domain is realized after passing to a suitable field.

A pre-symplectic left-symmetric algebra $\mathcal{A}$ is a left-symmetric algebra endowed with a nondegenerate skew-symmetric bilinear form $\omega$ such that all left multiplication operators are symmetric with respect to $\omega$. In this setting, the underlying subadjacent Lie algebra $(\mathcal{A}^{-},\omega)$ forms a flat $T$-symplectic Lie algebra. This paper provides a systematic investigation into the structural properties of pre-symplectic left-symmetric algebras. In particular, we introduce a distinguished subclass termed Milnor pre-symplectic algebras, and prove that any pre-symplectic left-symmetric algebra whose commutator ideal is nondegenerate necessarily belongs to this subclass. Next, we investigate the Levi-Civita product associated with symplectic Lie algebras. We show that this product always yields a right-symmetric algebra, and we prove that it forms a left-symmetric algebra if and only if it is associative. Furthermore, we provide a characterization of symplectic Lie algebras in terms of representations of left-symmetric algebras, and conclude by establishing a construction method for these structures known as the $T^*$-extension. Furthermore, we develop a double extension procedure for pre-symplectic left-symmetric algebras by means of commutative associative algebras. We show that every such algebra with a degenerate commutator ideal can be reconstructed via this extension process. More generally, we show that any pre-symplectic left-symmetric algebra is either a Milnor pre-symplectic algebra or can be obtained through a finite sequence of successive double extensions starting from a Milnor pre-symplectic algebra. As a concrete application of these structural results, we provide a complete classification of pre-symplectic left-symmetric algebras of dimension less than or equal to $4$.

We construct an $R$-algebraic analog of the Bunce-Deddens algebra using the theory of Leavitt labelled path algebras. In particular, we will construct a labelled space whose associated partial action on tight filters is exactly the odometer action on the Cantor set that induces the classical Bunce-Deddens algebra. After defining $R$-Bunce-Deddens algebras, we will prove $R$-algebraic analogs of various results for classical Bunce-Deddens algebras. As our primary application of these results, we will show that, for any field $K$, a $K$-Bunce-Deddens algebra is not Morita equivalent to any Leavitt path algebra.