Papers

For quandle rings, Bardakov–Passi–Singh and Elhamdadi–Nunez–Singh respectively formulated an analogue of Kaplansky's zero divisor conjecture and a weakened analogue of Kaplansky's idempotent conjecture involving the augmentation map. In this note, we verify these conjectures for medial quandles. The latter holds for a larger class of quandles; we characterize integral domains over which the statement holds for this class. We also characterize integral domains of characteristic 0 over which the statement holds for a given finite Alexander quandle.

Let $\mathbb{F}$ be a finite field and let $G$ be a group such that every finite subgroup of $G$ is cyclic. In this paper we classify the division gradings by cyclic groups on matrix algebras over $\mathbb{F}$. This yields the classification of the $G$-gradings on these algebras. Moreover if $G$ is abelian we prove that two matrix algebras over $\mathbb{F}$ with $G$-gradings satisfy the same graded identities if and only if they are isomorphic as graded algebras.

The Fomin-Kirillov algebras $\mathcal{FK}_n$ are shown to be infinite-dimensional for $n\geq 6$. By computer calculations, we find an explicit nonzero primitive element of degree 14 in the identity component of $\mathcal{FK}_6$. For every $n\geq 6$, the existence of such an element implies that $\mathcal{FK}_n$ contains a polynomial subalgebra and is therefore infinite-dimensional, and that the canonical homomorphism from $\mathcal{FK}_n$ onto its associated Nichols algebra is not an isomorphism.

Let $f,g\in\mathbb{Z}[t]$ be products of cyclotomic polynomials such that either $g/\gcd(f,g)$ or $f/\gcd(f,g)$ (or both) is squarefree. We give explicit formulae for the invariant factors of $f(C_g)$, where $C_g$ is the companion matrix of $g$, expressing them in terms of the indices and multiplicities of the cyclotomic factors of $f$ and $g$. A more compact formula holds for the circulant case $g(t)=t^n-1$, assuming that every cyclotomic factor shared by $f$ and $g$ has multiplicity $1$ in $f$. We discuss applications to several topics, including abelianizations of cyclically presented groups, homology groups of cyclic branched covers of knots, adjacency matrices and shifted graph Laplacians, $K$-theory, and Smith forms of matrix powers. For example, for two families of hyperbolic knots we give the first integral homology in every covering degree; or, when $g$ is squarefree, we determine the least eventual period of the Smith forms of powers of $f(C_g)$.

We present the complete classification of non-isomorphic complex $3$-dimensional unital commutative algebras. Every such algebra decomposes canonically as $\mathbb Ce_1\oplus V$, where $e_1$ is the unit and $V$ is the kernel of the trace form; the induced traceless $2$-dimensional commutative algebra on $V$ is one of four algebras, and the classification reduces to the classification of symmetric bilinear forms on $V$ up to the automorphisms of this kernel. The answer consists of six algebras, three one-parameter families, one two-parameter family and one three-parameter family.

The Okubo algebra brings together ternary gradings, integral root frames and rank-three geometry. We examine how these structures extend to the complex Exceptional Periodicity algebras. With $N=4(n+1)$, a tetracode character gives eight nonzero homogeneous components of dimension $t_n=8n+3+2^{4n}$. On a tip, a compatible grading has zero component of dimension $8n-5$ and eight nonzero components of dimension $1+2^{4n-3}$. The arithmetic behaves differently: for $n\geq2$, the lattice $D_N^+$ has no full-rank orthogonal frame of unscaled $A_2$ lattices. We exhibit failures of the Jacobi and Jordan identities and determine the derivations and full automorphism group of the ambient algebra above level one. We also give a Clifford-module proof, with the normalization fixed by root products, of the published realization of EP tips as special Vinberg $H$-algebras. This identifies a positive real form and yields explicit formulas for its cone, cubic, Cholesky coordinates and rational dual invariant. The cone is symmetric exactly at $n=1$. Its determinant-one hypersurface belongs to the known $L(8n,1,0)$ series, whose $r$- and $c$-maps give homogeneous scalar manifolds for supergravity theories with eight real supercharges.

Analogous to quasi-Frobenius rings, dual rings, and Johns rings, the classes of $p$-quasi-Frobenius rings, $p$-dual rings, $p$-Johns rings are introduced and used to discuss and settle principal analogues of the Faith-Menal, Faith, and FGF conjectures. A chain symmetry theorem establishes that for quasi-injective semi-projective modules with semiprime endomorphism ring, endonoetherianity and endoartinianity are equivalent and force the endomorphism ring to be semisimple artinian. Within this unified setting, it is shown that (1) two-sided $p$-Johns rings are $p$-quasi-Frobenius, giving a principal Faith-Menal analogue, with a counterexample showing that the one-sided hypothesis is insufficient; (2) semiprimary principal right ideal rings that are right principally injective and right Kasch are $p$-quasi-Frobenius-this affirmatively answers a question of Faith (Q2) in the principal two-sided ideal ring case and settles the principal Faith conjecture, with a counterexample showing the hypotheses to be indispensable; and (3) right $p$-Johns rings that are left CF and left perfect are $p$-quasi-Frobenius, yielding a principal FGF analogue, with a counterexample demonstrating that the hypotheses are essential. Wherever needed, examples are provided to show that various classes introduced are different from the existing classes in the literature.

This article surveys the main results on the representability of internal actions in varieties of non-associative algebras over a field. We begin by recalling the general framework of internal actions and split extensions in semi-abelian categories, before specialising to Orzech categories of interest and varieties of non-associative algebras. We discuss action representability, weak action representability, and action accessibility, with emphasis on the relationship between the latter two notions. We then describe the constructions of the universal strict general actor in Orzech categories of interest and the external weak actor in action accessible, operadic varieties of non-associative algebras over a field, as well as their role in the study of weak action representability. We also give a tensor-domain formulation of partial algebras and adapt the construction of the external weak actor to this setting. These notions are illustrated by examples arising from associative, Lie, Leibniz, Jacobi-Jordan, anti-commutative anti-associative, alternative, and $2$-nilpotent algebras. We conclude with a summary of the known results and some open questions for future research.

Let $W_n$ be the Lie algebra of polynomial vector fields in $n$ variables over a field of characteristic zero, and let $s_k$ denote the alternating sum of compositions of $k$ vector fields. The identity $s_{n^2+2n-1}=0$ has been asserted for arbitrary $n$. We give an explicit counterexample in dimension five, verified by an exact integer computation of a $34$-commutator. Together with the known vanishing of $s_{35}$, this shows that $35$ is the least degree of a standard identity on $W_5$. We then prove that $s_{n^2+2n-1}=0$ for every even $n\geq2$. The proof uses Lie algebra homology with coefficients in the polynomial algebra modulo constants and a lemma on invariant tensors. We describe the computation and provide the source code and verification results.

We introduce a strict monoidal groupoid associated with a given commutative ring $K$. It acts on the Clifford extension systems of $K$-central algebras which are graded by a group $G$, as well as on the generalized collective characters, both of type $\phi \in$ Hom$(G,$ Aut$(K))$. These actions partition the two sets above with respect to graded Morita equivalence. By that we establish a seven term exact sequence of type $\phi$, which boils down to the celebrated Chase-Harrison-Rosenberg sequence when $\phi$ is Galois.

A recent preprint constructs a finitely presented torsion-free group $G$ with a finite two-dimensional classifying space such that $\mathbb F_2[G]$ contains nonzero zero divisors. We extend this to every odd prime $p$: there is a finitely presented torsion-free group $G_p$, again with a finite two-dimensional classifying space, such that $\mathbb F_p[G_p]$ contains nonzero elements $\alpha,\beta$ with $\alpha\beta=0$. Together with the case $p=2$, Kaplansky's zero-divisor conjecture therefore fails over every field of positive characteristic. The new ingredients are a signed version of the parity cancellation, made possible by a bipartite random model on one side; a type design built from the projective plane over $\mathbb F_{p^3}$ with $p$ additional letters, which makes every cross-intersection congruent to $1$ modulo $p$; and a squared-word decay estimate for this design, proved with an explicit test vector. The geometric part of the argument is taken from the characteristic-two construction. The characteristic-zero case remains open.

We develop an effective method for determining the real Waring rank and the real border rank of biharmonic binary forms. By means of real apolarity, we convert both of these problems into membership questions for regions bounded by a hypocycloid and an epicycloid. We further provide a computable algebraic criterion for the relevant memberships, which involves only the signs of certain real-valued resultants.

Let $\varphi\colon A\to B$ be a Jordan homomorphism of associative algebras over a field $\mathbb{F}$ of characteristic different from $2$, with defect ideals $V_{\varphi}$ and $W_{\varphi}$. If $A=\operatorname{Alg}([[A,A],[A,A]])$, then $V_{\varphi}W_{\varphi}=W_{\varphi}V_{\varphi}=0$ and $V_{\varphi}\cap W_{\varphi}\subseteq\operatorname{Ann}(\operatorname{Alg}(\varphi(A)))$. Moreover, the induced map modulo $V_{\varphi}\cap W_{\varphi}$ is standard. The annihilator bound is sharp, even in finite dimension. As an application, when $A$ is unital and $A=K(A)$, every Jordan homomorphism defined on $A$ is splittable and standard. In this setting, the corresponding result of Brešar and Zelmanov holds without the surjectivity hypothesis.

Laurent phenomenon algebras generalise cluster algebras by allowing irreducible exchange polynomials in place of binomials. Lam and Pylyavskyy proved finite type for those arising from graphs and conjectured that the same holds for every Laurent phenomenon algebra admitting a linear seed. We prove this conjecture over an arbitrary unique factorisation domain. Our proof constructs a nonzero element of the upper algebra divisible by every cluster variable. Comparing noninitial cluster variables with the prime factors of this element in the initial Laurent ring bounds the number of cluster variable classes and yields a finite normalised mutation pattern.

The finite-dimensional $F$-algebras $D=F(e_{11}+e_{33})+Fe_{22}+Fe_{12}+Fe_{13}+Fe_{23} \subseteq UT_3$ and $D_0=Fe_{22}+Fe_{33}+Fe_{12}+Fe_{13}+Fe_{14}+Fe_{23}+Fe_{24}+Fe_{34} \subseteq UT_4$ occupy a distinguished position in the theory of central polynomials. Together with the infinite-dimensional Grassmann algebra, these algebras generate the only varieties of almost polynomial growth of proper central codimensions. We determine finite generating sets for their $T$-spaces of central polynomials and compute their central and proper central codimensions. We also prove that every group grading on $D_0$ is isomorphic to an elementary grading and determine the identities and codimensions associated with all its $\mathbb{Z}_2$-gradings.

We characterize the central extensions, in the sense of the categorical theory of central extensions of Janelidze and Kelly, relative to the Birkhoff subcategory of $n$-nilpotent objects of a semi-abelian and algebraically coherent category. An extension $f \colon A \to B$ with kernel $K$ turns out to be central precisely when the iterated Huq commutator $[A,[A,\dots,[A,K]\dots]]$, with $n$ occurrences of $A$, vanishes. For $n=2$ this condition reads $[A,[K,A]] = 0$, that is, $[K,A]$ is a central subobject of $A$. In the special case of groups this description was obtained by Everaert and the first author. The present proof is categorical, and applies in particular to Lie algebras, to crossed modules, to cocommutative Hopf algebras over a field, and to any category of interest.

Let \(\Lambda\) be a \(\Z\)-graded Artin algebra which is Iwanaga–Gorenstein. We prove that \(\Lambda\) is CM-finite if and only if the category of gradable Gorenstein-projective modules has finite type, and if and only if the lengths of the indecomposable gradable Gorenstein-projective modules are bounded. Moreover, if \(\Lambda\) is CM-finite, then every Gorenstein-projective \(\Lambda\)-module is gradable. These results provide Gorenstein-projective analogues of the corresponding theorems of Gordon–Green for graded Artin algebras of finite representation type.

We show that every proper variety of distributive $\ell$-pregroups is periodic; as a result, the subvariety lattice of the variety of distributive $\ell$-pregroups decomposes into a disjoint union of bounded intervals each consisting of all properly $n$-periodic varieties, for different $n \in \mathbb{Z}^+$. On the way, we study non-periodic (one-generated) subalgebras of $\mathbf{F}(\mathbb{Z})$ and show that they either contain a non-periodic idempotent or they contain idempotents of arbitrary big periodicity. The same results holds for non-periodic one-generated subalgebras of $\mathbf{F}(\mathbf{J} \overrightarrow{\times} \mathbb{Z})$, where $\mathbf{J}$ is a chain, generated by a composite element and the resulting idempotents can be taken to be local in the associated wreath product.

Generalizing the setting of the second author's work on indecomposable modules without nonzero hollow factors, we characterize quivers $Q$ for which the dual algebra $R$ of the path coalgebra of $Q$, equivalently the complete path algebra, has no nonzero uniform right ideals: this happens exactly when every vertex of $Q$ can reach a vertex emitting at least two distinct arrows. Equivalently, the injective hull $E(R)$ of the right regular module is superdecomposable. These conditions are also equivalent to the path coalgebra having no nonzero hollow factor modules as a left module over its dual algebra.

Membranes are sufficiently smooth vector-valued functions defined on the unit square, widely used in applications such as interpolation of image data. The signature of a membrane is a collection of nonlinear features that capture its essential geometry. We index these features in the graded Hopf algebra of two-parameter shuffle products and study the algebraic varieties for each graded component. At each truncation level, we obtain the dimension of this variety via a novel Witt-type formula. We propose a Chen-Chow theorem for membranes, provably at the second truncation level, and conjecturally for higher levels with extensive computational evidence in the computer algebra system OSCAR.

We develop a support theory for wide subcategories of essentially small Krull–Schmidt \(n\)-abelian categories. We construct spectra associated with families of wide subcategories and establish the corresponding support and radical theories. In particular, we characterize prime wide subcategories as the completely meet-irreducible elements of the lattice of wide subcategories and prove that every wide subcategory is recovered from the prime wide subcategories containing it. We further show that any classifying support datum is recovered, together with its object supports, from the canonical prime spectrum. Finally, we provide a \(2\)-abelian example admitting no classifying support datum.

Spectral duality structures (SDS) explain why a distinguished family of initial distributions makes the absorption probabilities of resetting processes insensitive to the resetting mechanism. Here we study their axioms. We prove that the six defining conditions are mutually independent and determine what each one contributes. The key tool is a structure theorem valid for an arbitrary permutation and an arbitrary positive weight: every insensitive distribution is supported on a union of cycles balanced at a common level, and the level of a cycle is fixed by the geometric mean of the weight on it. From it, positivity selects the admissible root, the constant orbit product is equivalent to the existence of an insensitive distribution of full support, and the involution is equivalent to the response rank being equal to the number of free weight parameters. Without the duality relation, insensitivity of a single distribution is cheap and generically isolated; duality turns it into structural neutrality of the separatrix. The full-rank condition splits into two layers: a weaker condition is equivalent to maximal response rank, while full rank also excludes insensitivity at any other level. We then show that a single SDS is internally rigid: compatible semigroup structures force a constant weight, compatible abelian group structures obey a precise arithmetic condition, congruences form a simple lattice, and the automorphism group is computed explicitly. Finally, the axioms do not determine the orientation of the response; in the two-site case we give the exact condition under which the geometric and analytic sign criteria agree.

Linear complementary dual (LCD) codes are linear codes that intersect their dual trivially. We study double circulant (Euclidean) LCD codes over the finite chain ring $R=\mathbb F_2+u\mathbb F_2$, where $u^2=0$. Using the Chinese Remainder Theorem and the factorization of $x^n-1$, we derive an explicit enumeration formula, with a separate treatment of the constituent corresponding to $x-1$, which is not obtained by a formal specialization of the previously known counting formula over $\mathbb F_q[u]/(u^k)$. Assuming Artin's primitive root conjecture for $2$, we show that the binary Gray images contain an asymptotically good family of rate $1/2$ whose relative distance can approach $H_2^{-1}(1/4)$ from below. More importantly from the constructive viewpoint, by combining Gray images with a known extension construction, we obtain a binary LCD code with parameters $[29,15,7]$ and construct two explicit permutation-inequivalent binary LCD codes with parameters $[45,23,9]$.

We prove the geometric halves of Hilbert's Nullstellensatz and of the real Nullstellensatz for submodules of finitely generated modules and for coherent sheaves, complementing our earlier result that semiprime (resp. real) submodules are intersections of prime (resp. real prime) submodules. First, we pass from prime to maximal submodules: over a Jacobson ring, a prime submodule $P$ of $M$ is an intersection of maximal submodules whenever $M/P$ is a torsionless $R/(P:M)$-module, in particular whenever $M/P$ is finitely generated (the latter case is due to Maani Shirazi and Sharif). Consequently, if $R$ is a finitely generated algebra over an algebraically closed field and $M$ is finitely generated, the smallest semiprime submodule containing a submodule $N$ of $M$ consists of all $f\in M$ such that, for every point $a$, the value $f(a)$ (i.e. the image of $f$ in the fibre $M/\mathfrak{m}_aM$) lies in $N(a)$ (i.e. the image of $N$ in $M/\mathfrak{m}_aM$). We prove the real analogues over real Jacobson rings; for finitely generated algebras over a real closed field they are determined by the real points. The module classifications and directional vanishing formulas yield both Nullstellensätze for coherent sheaves on schemes of finite type over a field. We also prove compatibility of semiprimeness, reality and both radicals with arbitrary localization, and the resulting stalk criteria.

Let $R$ be a commutative ring. A submodule $N$ of an $R$-module $M$ is semiprime if $m\in(N:m)M$ implies $m\in N$. It is known that every semiprime submodule of $R^n$ is an intersection of prime submodules, and Aryapoor extended this to finitely generated modules. We show that no finiteness assumption is needed: every semiprime submodule of an arbitrary $R$-module is the intersection of the prime submodules containing it. The proof is short and uses neither coordinates nor presentations; its key step is that a nonzero element $q$ of a semiprime module $Q$ satisfies $sq\notin\p Q$ for all $s\notin\p$, where $\p$ is any minimal prime over the annihilator of $q$. We define an $R$-module $M$ to be real if $\sum_i m_i\otimes m_i=0$ in $M\otimes_RM$ implies $m_i=0$ for all $i$, and a submodule $N\subseteq M$ real if the module $M/N$ is real. We prove the corresponding result for real submodules: every real submodule of an arbitrary module is the intersection of the real prime submodules containing it. On $R^n$ our notion of realness coincides with the coordinate definition used in earlier work on real Nullstellensätze for free modules.

We initiate a systematic operadic approach to anti-pre-Lie algebras. We study the operad controlling anti-pre-Lie algebras, its relationship to the operad of Novikov algebras and its cyclic structure. We establish that the Lie bracket of an anti-pre-Lie algebra satisfies the standard identity of degree five. We also establish a quadratic maximality property of the anti-pre-Lie operad with respect to this identity. We then show that the anti-pre-Lie operad and its Koszul dual are not Koszul and give a full description of the latter. Finally, we identify the Lie elements in the Koszul dual operad with the operad of metabelian Lie algebras.

Let $F$ be a field of positive characteristic, and let $A$ be a finitely generated commutative associative $F$-algebra. We study Lie subalgebras of $Der \, A$ generated by locally finite derivations. We prove that if all commutators of the generators satisfy a natural local finiteness condition, then the associative subalgebra of $End_{F}\, A$ generated by these derivations is finite-dimensional.

We consider a family $(\Lambda_\lambda)_{\lambda \in \Bbbk^\times}$ of weakly symmetric algebras of Euclidean type over an algebraically closed field $\Bbbk$, where $\Bbbk^\times:= \Bbbk\setminus \{0\}$, and for each $\lambda \in \Bbbk^\times$, $\Lambda_\lambda$ is socle equivalent to $\Lambda_1$, which is the trivial extension of a hereditary algebra of type $\tilde{A}$. We compute their Hochschild cohomologies, which shows that $\Lambda_\lambda$ is not derived equivalent to $\Lambda_1$ if $\lambda \neq 1$. This means that a derived equivalence classification of weakly symmetric algebras of Euclidean type given by Bocian–Holm–Skowroński is not complete.

We prove that a Hopf algebra over an arbitrary field has a bijective antipode if either its left or right regular module has finite uniform dimension. In particular, this solves Skryabin's conjecture, posed in 2006, that every noetherian Hopf algebra has a bijective antipode. Starting from Skryabin's injectivity theorem for weakly finite Hopf algebras, we associate to each finite-dimensional subcoalgebra an essential left ideal of common denominators that is also a right coideal in the antipode image. An annihilator identity makes this image essential in the original algebra as a left module over the image, yielding finite left uniform dimension from finite right uniform dimension. A second denominator construction inside the antipode image, together with a restricted inverse-antipode identity, proves surjectivity.

Emil Artin asked whether the left and right dimensions of an extension of skew fields must agree. Cohn and Schofield gave counterexamples through elaborate constructions. It has been a long-standing open question whether there is any division ring, infinite-dimensional over its center, for which the left and right indices agree whenever at least one is finite. We give the first such examples. Call a division ring index-symmetric if every division subring with at least one finite index has equal finite indices. Using a duality formula for bimodules over division algebras, we prove that every twisted Calabi–Yau division algebra is index-symmetric. The examples include the Weyl skew fields and, more generally, the quotient division rings of rings of differential operators on smooth affine varieties in characteristic zero. Further examples are the quotient division rings of enveloping algebras of finite-dimensional Lie algebras, Ore extensions of twisted Calabi–Yau noetherian domains, quantum affine spaces, and group algebras of torsion-free polycyclic-by-finite groups.