Papers

Using the duality between the categories of inverse semigroups and ample groupoids, we prove the triviality (global and local) of the central term of the extension in terms of twisted crossed products.

Non-commutative friezes have been introduced by Berenstein and Retakh and studied further by the authors. In this paper we consider non-commutative friezes over normed division rings, like for instance Hamilton's quaternions or more general quaternion algebras. We address the fundamental question in the theory of friezes of whether over a certain subset there are finitely or infinitely many non-commutative friezes (with 1's on the boundary) for any height. As an application of a theorem bounding the norm of quiddity entries we deduce that for every norm-finite subset of a normed division ring there are only finitely many such non-commutative friezes for every height. In particular this result applies to the Lipschitz quaternions and the Hurwitz quaternions of Hamilton's quaternions. We then study more generally non-commutative friezes over Lipschitz subrings of non-split quaternion algebras $(a,b)_{\mathbb{Q}}$. We determine the frieze subrings for all $a,b<0$, and as a consequence we see that all such non-commutative friezes are known if $a\le -4$ and $b\le -4$.

We give a systematic description of $\Hom$-Lie structures on several classes of Lie algebras and of their relations with transposed Poisson structures. Our starting point is the observation that the left-hand side of the defining identity is an alternating trilinear map; this makes $\mathcal{H}(\LL)$, the set of $\Hom$-Lie structures of $\LL$, a subspace of $\End(\LL)$ and reduces its computation to increasing triples of basis vectors. From this we derive, in every finite dimension $n$, the lower bound $\dim \mathcal{H}(\LL)\geq n \dim \mathcal{C}(\LL)$ and the codimension bound $\binom{n}{3}\dim [\LL,[\LL,\LL]]$, together with a refinement of the latter, and we determine $\mathcal{H}(\LL)$ for nilpotent, oscillator and solvable algebras, for $\sltwo \oplus A_{n-3}$, for the Witt, Virasoro and Heisenberg–Virasoro algebras, for all three-dimensional algebras, for $M_n(\bF)$ and $T_n(\bF)$, for the Virasoro-like algebra and its $q$-analogue, and for the planar Galilean conformal algebra. By a result of Filippov, every $\frac12$-derivation, and more generally every $\delta$-derivation with $\delta\neq0,1$, of a Lie algebra is a $\Hom$-Lie structure; we observe that, more generally, for $\delta\neq 0,1$ the $\delta$-derivations of an algebra of any variety defined by identities of degree at most three (associative, Novikov, left-symmetric, Leibniz, \ldots) are $\Hom$-structures of the corresponding $\Hom$-variety. We show that the inclusion of the $\frac12$-derivations into the $\Hom$-Lie structures is far from being an equality by computing the graded $\Hom$-Lie structures of Witt-, Virasoro- and Schrödinger-type algebras and of their central extensions, for which we give an explicit lifting obstruction. Along the way we correct two multiplication tables from the literature that violate the Jacobi identity.

We classify the simultaneous half-bandwidths \(p,q\) of a symmetric oscillatory matrix \(A\) and its inverse. At order \(n\ge2\), the possible pairs are \((1,n-1)\), \((n-1,1)\), and all pairs with \(2\le p,q\le n-1\). Every pair has an integer realization of determinant one. Let \(e_k(A)\) be the least positive integer \(m\) for which all minors of order \(k\) of \(A^m\) are positive. We prove \[ e_k(A)\le \min\{\max\{k,n-p\},\max\{n-k,n-q\}\}\qquad(1\le k<n). \] When \(p+q\ge n\), one realization attains all these bounds simultaneously. The maximum of \(\sum_{k=1}^{n-1}e_k(A)\) is determined for every bandwidth pair. Write \(A=U^T\Delta U\), with \(U\) unit upper triangular and \(\Delta\) positive diagonal. Let \(\ell(U)\) be the least number of positive elementary upper bidiagonal factors of \(U\). At fixed \((p,q)\), its minimum is \(\max\{n-1,p+q-1\}\). The bound \(\sum_{k=1}^{n-1}e_k(A)+\ell(U)\le n(n-1)/2+n-1\) is sharp at each factor count; equality is characterized using \(U\), \(U^2\) and the zero corner minors of \(A\). In particular, \(\sum_{k=1}^{n-1}e_k(A)\le n(n-1)/2\), with equality exactly for basic matrices. In the basic oscillatory class, without a symmetry assumption, a positive integer vector \((f_1,\ldots,f_{n-1})\) is realizable if and only if \(f_k\le\max\{k,n-k\}\) and \(f_k+f_{n-k}\ge n\) for \(1\le k<n\), and \(\lvert f_{k+1}-f_k\rvert\le1\) for \(1\le k<n-1\). Explicit factor orders realize every such vector.

By definition, a dual reflection group is a finite group which grades an Artin–Schelter regular algebra $A$ such that the identity component $A_e$ is again regular. In this paper, we completely classify the nonabelian dual reflection groups for regular algebras of dimension $3$: we first classify all possible nonabelian gradings which refine the natural $\mathbb{N}$-grading of the algebra, and then determine which of these gradings yield dual reflection groups. We show that none of the cubic algebras admit dual reflection groups, while for quadratic algebras, the dual reflection groups arise either from Ore extensions of two-dimensional regular algebras, or from the Sklyanin algebras $S_{q,0,1}$. In the Sklyanin case, the dual reflection groups form a new infinite family of groups $\Delta_n$ of order $27n^3$, answering a question of Goetz, Kirkman, Moore, and Vashaw. We also extend the four-dimensional examples of the same authors to two new infinite families of dual reflection groups. Finally, we show that an AS regular algebra admitting a dual reflection group need not have binomial relations, answering another of their questions. However, we conjecture that the relations must be binomial when $A_e$ contains no elements of degree $1$, and we prove this for nonabelian gradings in dimension $3$.

In this paper, we give conditions on a unital non-associative ring $R$ and additive map $\sigma: R\longrightarrow R$ so that the non-associative skew power series ring, denoted by $R[[X;\sigma]]$ and the non-associative skew Laurent series ring denoted by $R((X;\sigma))$ are left (resp. right) Noetherian, respectively. Consequently, we establish versions of Hilbert Basis Theorem of $R[[X;\sigma]]$ and $R((X;\sigma))$, respectively, for left (resp. right) Noetherianity . We also give a counterexample showing that the Hilbert Basis Theorem does not hold in general in the non-associative case. In establishing the Hilbert Basis Theorems for $R[[X;\sigma]]$ and $R((X;\sigma))$, we introduce and utilize the right (resp. left) quasi-associative property of $R$, which is equivalent to $rR$ (resp. $Rc$) being one-sided ideals for all $r\in R$ (resp. $c\in R$).

We introduce Boolean trusses, that is, trusses in which every element is idempotent. Every Boolean truss admits a decomposition into a truss arising from a Boolean ring and a rectangular truss. Furthermore, we classify all Boolean trusses up to isomorphism.

The study of preordered semirings is motivated by applications in computer science, graph theory, and information theory, and provides tools for understanding the asymptotic preorder, which compares large powers of a pair of elements. This paper studies sequences which behave approximately as sequences of powers, but are not necessarily equivalent to geometric sequences. Our main result is that preordered semirings admit completions where such sequences, that we call approximately geometric, become equivalent to geometric sequences, and that existing characterizations of the asymptotic preorder extend to the completion. We provide several classes of examples of approximately geometric sequences in the semiring of tensors, and in the semiring of graphs. As a concrete application, we determine the strong converse exponent for binary hypothesis testing with composite Markov hypotheses.

We give explicit algorithms that factor elements of the standard diagram monoids into their usual generators. These algorithms generalize sorting from permutations to partial matchings and set partitions. In every case the worst-case complexity is $n^2$, which is optimal for algorithms that explicitly list the factors. We also determine the average complexity of our algorithms.

In the year $1977,$ while developing the theory of modules arising in the context of Serre's problem on projective modules in the paper “On Stably Free Modules" (see \cite{AS}), A. A. Suslin constructed matrices inductively using unimodular rows. Our aim is to connect these matrices to the current and applicable field of coding theory. To this end, we construct Suslin-Rao codes, by using looking at certain unimodular all one vectors over the finite field $\mathbb{F}_2$. Comparative analysis with standard families such as Hamming, Simplex and BCH codes shows that the Suslin-Rao codes achieve better performance than Hamming and Simplex codes in terms of code rate and error correction capability, though BCH codes may outperform them for large code lengths. Bit error rate (BER) versus signal-to-noise ratio (SNR) simulations further support the practical applicability of these codes in noisy channels. The recursive nature of the construction allows efficient encoder design, making the proposed codes attractive for real-time communication systems and hardware implementation.

We associate to finite block size data \[ \bm n=(n_d)_{d\ge1},\qquad n_1>0, \] a positive definite cyclotomic form and a Householder reflection representation. Writing \[ N=\operatorname{lcm}\{d:n_d\ne0\},\qquad \wt=\sum_d d^2n_d, \] the form on the cyclic group $C_N$ is \[ B_{\bm n}(g,h)=\frac1{\wt}\sum_{(g^{-1}h)^d=1}d^2n_d. \] Its labelled Fourier spectrum reconstructs the complete multiplicity sequence by Möbius inversion. If the $n_d$ are the Wedderburn multiplicities of a split semisimple algebra $S=\prod_d M_d(k)^{n_d}$, the divisibility widths are dimensions of characteristic sums of Wedderburn blocks, and the form together with $\dim_kS$ recovers the isomorphism type of $S$. The main theorem is an arithmetic rigidity result. For $N\ge2$, the reflection group is finite if and only if it preserves a full rank lattice, equivalently \[ \supp(\bm n)=\{1,N\},\qquad n_1=N^2n_N. \] In this case $B_{\bm n}=\frac12(I+J)$ and the group is $W(A_N)\cong S_{N+1}$. In semisimple algebra terms, finiteness is equivalent to \[ S\cong k^{N^2m}\times M_N(k)^m \] for some $m\ge1$. For $N=2$, every nonfinite case is a dense copy of $D_\infty$ in the corresponding orthogonal group. For contraction algebras of threefold flops, Toda's width formulas identify the total scale and primitive Fourier mode with $\dim A_{\con}$ and $\dim A_{\con}^{\ab}$. This yields infinitude for all irreducible flops of length at least three and, in length two, \[ \Gamma_f\text{ finite}\quad\Longleftrightarrow\quad \dim A_{\con}=2\dim A_{\con}^{\ab}. \] The final part gives a complementary BPS/Riemann–Hilbert realization of the same reflection matrices.

We prove the Fraser–Sherman-Bennett conjecture on the quasi-equivalence of positroid cluster structures arising from admissibly relabeled plabic graphs. We construct a tilting module from the boundary necklace and obtain a Frobenius categorification of the relabeled cluster structure. By identifying localized characters of rank-one modules with Plücker coordinates, we match the stable images of the mutable face modules with the summands of a reachable cluster-tilting object in the stable category. The induced derived equivalence yields a unimodular transformation of relative index lattices intertwining the full extended exchange matrices. These comparisons show that the identity map on the coordinate ring is a quasi-cluster isomorphism between the two cluster structures.

We study the minimal dimension of a nonsingular anisotropic universal quadratic form over a field $F$ of $\operatorname{char}(F)=2$, which we call the “$m$-invariant" of $F$. We show that every even number is the $m$-invariant of some field of characteristic 2, and consider cases when $u(F)$ and $m(F)$ are equal and when they are not.

The well-known Hua identity $a - (a^{-1} + (b^{-1} - a)^{-1})^{-1} = aba$ holds in any associative division ring. We prove, in a sense, a converse: if a (nonassociative) ring with unit and multiplicative inverses satisfies the Hua identity, then it is alternative.

We prove Laurent positivity for skew-symmetrizable quantum cluster algebras of rank three, with arbitrary integral compatible quantizations and invertible frozen variables. The proof uses mutation-acyclic positivity from a companion manuscript.

We introduce a new structural framework for constructing multivariate Alexander quandles based on a groupoid ${\cal G}$, composed of a disjoint union of delooping groupoids of deck groups. Compared to standard Alexander modules over group rings ${\mathbb{Z}}[{\mathbb{Z}}]$, the multivariate version involves modules over groupoid rings ${\mathbb{Z}}[{\cal G}]$. A manifestly categorical description is given in terms of ${\mathbb{Z}}$-Algebroids, whose functorial module theory yields an equivalent definition of multivariate Alexander modules. Using the category of elements construction, we define multivariate Alexander quandles from this groupoid framework. Three classes of multivariate quandle operations follow. In contrast to earlier approaches, we abstract the algebraic structure from explicit link-dependence. As a consequence, we have a structural framework for: (i) enumerating new classes of Alexander quandles, and (ii) finding new coloring invariants of links. We show that maps between ${\mathbb{Z}}$-modules, facilitated via multivariate quandle operations, realize a quiver presentation of quandles, which makes manifest the oidification of Alexander modules. Oidified Alexander modules, based on groupoids, provide a constructive framework for composing univariate quandles to obtain new multivariate ones. Finally, we comment on the possibility of new link invariants coming from groupoid-based formulations, including the fundamental groupoid.

Let $\mathcal G=H\rtimes\Gamma$ be a semidirect product of a finite group $H$ with $\Gamma=\mathbb Z_p^d$, the direct product of finitely many copies of the additive group of the $p$-adic integers. In the case $d=1$, Iwasawa algebras of such profinite groups were studied by Ritter–Weiss, Lau, Nickel, and the author. We generalise these results to all $d\ge1$, describing the Wedderburn decomposition of the total ring of quotients of the Iwasawa algebra of $\mathcal G$.

In this paper, we study regularity and clean-type decompositions for matrices over dual numbers. We characterize the von Neumann regularity of dual matrices by establishing a necessary and sufficient compatibility condition between their real and dual components. We determine the conditions under which a dual matrix $\mathcal{M} = A + B\epsilon$ is regular, $\pi$-regular, nil-clean, or strongly nil-clean. Using the solvability of Sylvester matrix equations, we show that the matrix ring over dual numbers $M_n(\mathbb{D})$ is strongly $\pi$-regular and strongly clean. Moreover, $M_n(\mathbb{D})$ is $\pi$-regular, clean, $r$-clean, strongly $r$-clean, $\mathrm{NR}$-clean, and strongly $\mathrm{NR}$-clean. On the other hand, $M_n(\mathbb{D})$ fails to be regular, strongly regular, nil-clean, strongly nil-clean and uniquely $\mathrm{NR}$-clean, although it inherits several strong decomposition properties from $M_n(\mathbb{R})$.

We present a new proof of a classical theorem of Dieudonné: if a linear space of $n\times n$ matrices consists entirely of singular matrices, then its dimension is at most $n^2-n$. Our proof is based on a surprising “duality” argument: we prove this universal upper bound by exhibiting a single matrix space that serves as a lower bound for a related problem. Interestingly, this approach only works for certain fields, but we use model-theoretic arguments to obtain the same result for all fields. We hope that this approach can be generalized to provide new duality-based proofs of other classical theorems on matrix spaces, and give some preliminary results in this direction.

Jiao and Ren proved that the varieties generated by the matrix semirings $\M_n(S_7)$ form an ascending chain of nonfinitely based varieties, and asked whether consecutive members are equal for all $n\geq2$. We give a negative answer by exhibiting an identity in five variables that holds in $\M_2(S_7)$ and fails in $\M_3(S_7)$. Consequently, $\V(\M_2(S_7))$ is a proper subvariety of $\V(\M_n(S_7))$ for every $n\geq3$. The proof uses the cubic-power property established by Jiao and Ren and a description of products equal to the constant matrix $[a]_n$. The identity forces two distinct indices to exhaust the index set in dimension two; an additional index yields an explicit counterexample in dimension three.

We introduce and study the class of almost subtractive submonoids of additive monoids. This property relaxes the strict condition of subtractivity while still allowing for a simulated subtraction operation sufficient for many algebraic applications. We explore the structural properties of these monoids and demonstrate their significance in the theory of modules over semirings, specifically regarding quadratic forms and their companions.

We study braidings and infinitesimal braidings on categories of bi(co)modules and tetramodules, and their compatibility with duoidal structures. For an arbitrary coalgebra, we classify braidings on its category of bicomodules with the cotensor product in terms of canonical $R$-forms, dualizing the classification for bimodules obtained by Agore, Caenepeel, and Militaru. Every such braiding is a symmetry and admits only the zero infinitesimal braiding. We construct duoidal structures on categories of bi(co)modules and tetramodules over bimonoids in braided monoidal categories under suitable assumptions on equalizers and coequalizers. We then introduce pre-Cartier duoidal categories and their one-sided variants. We establish obstructions to compatible braidings for tetramodules, construct nonzero one-sided pre-Cartier structures on bi(co)modules, and give a pre-Cartier duoidal example with distinct monoidal products and a nonzero infinitesimal braiding.

We report an explicit construction of a 27 x 27 x 27 symmetric tensor with rational entries that has tensor rank 55 over the rational numbers and symmetric tensor rank 56 over the complex numbers, providing a small counterexample to Comon's conjecture over the rational, real, and complex numbers. The construction follows the framework of symmetric adjoins introduced by Shitov. The primary technical contribution of this work is to prove a special case of Conjecture 6 appearing in Shitov's seminal 2018 work.

We introduce the notion of factorizable Lie conformal bialgebras and establish their correspondence with quadratic Rota-Baxter Lie conformal algebras of nonzero weight. Consequently, on the one hand, a Lie conformal bialgebra with a Rota-Baxter operator of nonzero weight gives a factorizable Lie conformal bialgebra. And on the other hand, as the conformal analogue of the fact that a quadratic Rota-Baxter Lie algebra induces a generalized pseudo-Hessian post-Lie algebra and a special partial-pre-post-Lie algebra, a quadratic Rota-Baxter Lie conformal algebra as well as a factorizable Lie conformal bialgebra induces a generalized pseudo-Hessian post-Lie conformal algebra and a special partial-pre-post-Lie conformal algebra. Furthermore, we generalize the correspondence between Gel'fand-Dorfman bialgebras and a class of Lie conformal bialgebras to the factorizable cases. There is not only a correspondence between factorizable Gel'fand-Dorfman bialgebras and a class of factorizable Lie conformal bialgebras, but also a correspondence for their corresponding quadratic Rota-Baxter counterparts as well as some induced structures. In particular, there is a construction of factorizable Lie conformal bialgebras from Gel'fand-Dorfman bialgebras with a Rota-Baxter operator of nonzero weight.

Under suitable group actions, it is known that $A_\infty$-structures descend to orbit categories. We study the converse problem of lifting an $A_\infty$-structure to a prescribed covering. We show that a compatible group grading determines a canonical lift, uniquely characterized by the strictness of the covering projection. As an application, we construct $A_\infty$-categories associated with admissible fractional Brauer graph algebras by lifting Brauer graph $A_\infty$-categories. Their geometric data augment the surface models of Brauer graph algebras with a Nakayama character encoding the covering. We prove that admissible fractional Brauer graph algebras with equivalent geometric data are derived equivalent. Furthermore, we establish a set of combinatorial derived invariants and prove their completeness in reduced genus zero and in reduced genus at least two when the defining ribbon graph is non-bipartite.

We give a unified treatment of definite Albert algebras over $\mathbb{Z}$, with exceptional Siegel–Weil identities as the main arithmetic tool. Starting from the standard octonion order alone, its first scalar theta coefficient saturates the genus average and forces every remaining class to be rootless. A cubic-ring coefficient of the $G_2$ theta lift then gives the mass of the remaining scale-$2$ framed orders, also recovered from shell geometry. A uniform Niemeier–Golay stabilizer bound saturates this mass, proving that there is exactly one further class and determining both automorphism orders, frame transitivity, and the $2457$ frames on $819$ rank-one elements. The associated generalized hexagon identifies the nonstandard group as ${}^3D_4(2):3$ only after its order has been proved. An independent weighted-theta argument gives the trace-zero root-system alternative $\varnothing$ or $A_2E_8^3$. The construction relates positive masses of framed objects to exceptional Eisenstein coefficients.

We construct a graded Lie algebra structure on the space of cyclically skew-symmetric cochains whose Maurer-Cartan elements characterize double Lie algebra structures. Twisting by a fixed double Lie bracket yields a cochain complex that is isomorphic to the positive arity part of the complex introduced by Fairon and Valeri after reindexing and degreewise sign adjustment. Using the Koszul resolution of the double Lie properad, we identify the resulting differential graded Lie algebra with the convolution differential graded Lie algebra on the deformation complex of the fixed double Lie algebra. We also construct a graded Lie algebra governing skew-symmetric Rota-Baxter operators on a finite-dimensional symmetric Frobenius algebra. For a finite-dimensional vector space $V$ and $A=\en(V)$, we lift the known correspondence between double Lie algebra structures on $V$ and skew-symmetric Rota-Baxter operators on $A$ to an isomorphism of their governing graded Lie algebras. After twisting by corresponding Maurer-Cartan elements, this isomorphism becomes an isomorphism of differential graded Lie algebras and hence induces an isomorphism between the associated cohomology groups. We further prove that the cohomology of a skew-symmetric Rota-Baxter operator on a symmetric Frobenius algebra can be seen as the cyclic cohomology of its descendent associative algebra. Thus the cohomology of a double Lie algebra is identified with the cyclic cohomology of the descendent associative algebra associated with the skew-symmetric Rota-Baxter operator on the matrix algebra.

We study how determinant recurrences of banded Toeplitz matrices behave when their Laurent symbols are multiplied. Two classical structures underlie the problem. Clean banded Toeplitz determinants have a root-product description of their recurrences, while a product of two finite Toeplitz sections differs from the finite section of the product by finite-rank corner corrections. We connect these structures by identifying multiplication-induced boundary defects with exterior-degree sectors of the determinant recurrence. To each polynomial core we associate recurrence polynomials for all exterior degrees. Multiplication of cores becomes a graded convolution via composed products, while complementary degrees are related by a scaled reciprocal duality. On the matrix side, one-sided factors produce a defect filtration whose successive pieces are exactly these sectors. For general two-sided factors, independently weighting the two corner corrections gives an interval filtration: increasing either defect order adds one adjacent sector, and the cumulative recurrences are generically minimal. The multiplication-induced same-corner minors used in these filtrations lie in the standard row-column state module, whereas cross-corner imbalance corresponds instead to Laurent recentering. Opposite Laurent recenterings realize the individual sectors directly, and for several factors the admissible recenterings form an explicit lattice polytope. The pentadiagonal case gives the basic $1+4+1$ decomposition. The resulting framework gives a factor-level description of how finite-section boundary effects generate the recurrence spectrum of structured banded Toeplitz products.

We construct infinitely many pairwise non-derived-Morita-equivalent phantom categories $\mathcal{P}_{m}, m\in\mathbb{Z}$ from concrete directed DG quivers with 3 vertices. Moreover, there exists a fixed finite-dimensional DG algebra $C$ such that for each $m\in\mathbb{Z}$, there is a way to reassign cohomological gradings on $C$ (without changing the differential) to obtain a new DG algebra $C_{m}$ with $\mathrm{Perf}(C_{m})\cong \mathcal{P}_{m}$.

We prove that the three-element additively idempotent semiring $S_7$ is strongly nonfinitely based: every finite additively idempotent semiring whose generated variety contains $S_7$ has no finite equational basis. This answers Problem~1.2 in Yue, Ren and Gao's paper Explicit equational bases for the power semirings of $S_7$ affirmatively. The proof combines a rigidity property of set systems with private points and a probabilistic construction derived from semilinear sets of letter counts in finite semigroups. The argument allows arbitrary polynomial substitutions and does not assume commutative multiplication. As consequences, both power semirings of the multiplicative reduct of $S_7$, and both power semirings of every finite group of order at least three, are strongly nonfinitely based. All algebras are considered in the signature $(+,\cdot)$ without nullary operations.