Papers

We construct, over every formally real field, a singular pair of quadratic forms in dimension seven whose adjoint closure equals its pencil and consists entirely of hyperbolic forms in First's sense, although the pair is not weakly hyperbolic. This disproves First's conjecture that his local–global criterion for nonsingular pairs extends to singular pairs. The example has minimal dimension over formally real number fields and real closed fields. The construction uses an explicit closure formula: adjoining a symmetric singular Kronecker block of positive minimal index to a pair with a nondegenerate first member forces the adjoint closure of the sum to equal its pencil. We determine the corresponding closure formulas for Kronecker decompositions and compute the adjoint algebra, its Jacobson radical, and the involution-trace form of the seven-dimensional example.

Audenaert introduced a class of complementary McCarthy type trace inequalities in his work on completely monotone functions and Bernstein functions; the same problem was later included in the problem list of Audenaert and Kittaneh. The known results cover the corresponding directions for $q\le -2$, $0<q\le 1$, $1\le q\le 2$, and $2\le q\le 3$, while the negative range $-2<q<0$ was left as a conjectural case. We prove the negative exponent inequality, in fact for every $q<0$. After the inversion $X=A^{-1}$, $Y=B^{-1}$, the problem reduces to a trace inequality for the parallel sum $X:Y$. The proof uses the Kubo–Ando mean chain, an Ando–Hiai type log-majorization for matrix geometric means, and a finite-dimensional Schatten Hölder inequality, including the quasi-norm range. We also record that the positive exponent side $q\ge 3$ follows directly from Audenaert's norm-compression inequality for positive semidefinite $2\times2$ block matrices, and in fact holds for all $q\ge 2$. Finally, we determine the equality case on the negative side: equality holds if and only if $A=B$.

Let $R$ be a finite commutative local principal ring. We study products of nilpotent and idempotent matrices in $M_2(R)$. We show that every product of nilpotent and idempotent matrices is either a product of idempotents or a product of nilpotents. We then consider IN- and NI-matrices, that is, matrices which can be written as a product of an idempotent and a nilpotent matrix in the respective orders. We prove that the classes of IN- and NI-matrices in $M_2(R)$ coincide and give an explicit description of their common class. Finally, if $|R|=q^n$ and $R/J(R)\cong GF(q)$, we show that \[ |\operatorname{IN}(M_2(R))| = |\operatorname{NI}(M_2(R))| = q^{3n-2}(q^n+q^2-1). \]

Let $H$ be a Hopf algebra with a bijective antipode over a field $\mathbf k$ and $B/A$ be a flat right $H$-Galois extension. Ştefan constructed a spectral sequence converging to the Hochschild cohomology $\mathrm{HH}^{p+q}(B, N)$ with $\mathrm{E}_2^{p,q} = \mathrm{H}^p(H, \mathrm{HH}^q(A, N))$. We show that when $B/A$ is faithfully flat, the Ştefan spectral sequence is multiplicative. More precisely, a new formalism of functorial multiplicative spectral sequences via lax monoidal functors over monoidal categories is introduced. A functorial multiplicative spectral sequence is constructed for any faithfully flat Hopf Galois extension. Identification with Ştefan's spectral sequence from the $E_2$-page follows from Künzer's comparison theorems. Applications include strongly graded algebras, smash products, crossed products, group and Lie algebra extensions, with new multiplicativity results in several cases.

In this note, we prove that, under Kabbaj's weighted polynomial Hilbert-series hypothesis, every $N$-Koszul graded algebra of finite global dimension with $N>2$ is necessarily $3$-Koszul of global dimension $3$.

We give negative answers to two tilting-completion questions for finite-dimensional algebras. We construct two finite-dimensional basic connected quasi-hereditary $\mathbb C$-algebras. The first admits a faithful basic full-rank pretilting module with no tilting completion; the second admits an almost-tilting module with no tilting completion. The full-rank example also yields counterexamples to two conjectures: Enomoto's Self-orthogonal Wakamatsu-tilting Conjecture and the Self-orthogonal Faithful Conjecture of Chen, Li, Zhang, and Zhao. We further show that the Self-orthogonal Wakamatsu-tilting Conjecture holds for all finite-dimensional algebras if and only if the Self-orthogonal Faithful Conjecture holds for all finite-dimensional algebras. The construction ultimately stems from Krah's non-full exceptional collection of maximal length on a rational surface and Kalck's associated full-rank presilting example.

Let \(\F\) be an algebraically closed field of characteristic zero, \(\g=\s\oplus\z\) a finite-dimensional reductive Lie algebra over \(\F\), and \(V\) an arbitrary finite-dimensional \(\g\)-module. We classify all 2-local derivations of \(\g\) on \(V\), and show that every 2-local derivation is a derivation if and only if \(\dim\z\leq1\) or \(V^\g=0\). If \(\dim\z\geq2\) and \(V^\g\ne0\), the nonlinear homogeneous maps give all exceptional 2-local derivations.

For normal matrices A and B, the classical Hoffman-Wielandt theorem bounds the optimal matching distance between their spectra by the Frobenius norm of A-B. We prove the sharper estimate d(\sigma(A),\sigma(B))^2 \le \sum_{k=1}^{\lfloor (n+1)/2 \rfloor} s_k(A-B)^2, where s_k(M) are the singular values of M, involving only the first \lfloor (n+1)/2 \rfloor singular values of the difference. Consequently, d(\sigma(A),\sigma(B)) \le \sqrt{\lfloor (n+1)/2 \rfloor}\,\|A-B\|, which improves the classical dimension-free bound for all 3 \le n \le 16. The proof combines a min-max duality for optimal matching distances with spectral subspace overlaps and the monotonicity of singular values under rectangular compressions.

Examples and counterexamples define the boundaries of mathematical propositions, test foundational conjectures, and resolve open structural problems in abstract algebra. In loop theory, an Osborn loop is a crucial generalization of a Moufang loop satisfying a specialized variable-sandwiching identity. While small finite non-associative examples are known, constructing large examples of proper Osborn loops (loops that are neither conjugacy closed nor Moufang) remains a computational bottleneck. In this paper, we establish an efficient framework for generating large proper Osborn loops by taking the direct product of non-associative conjugacy closed (CC) loops and Moufang loops. We outline the boundary constraints necessary to prevent structural collapse into sub-varieties and provide explicit examples up to order 2025 validated via the GAP package LOOPS.

Using determinantal divisors of integral matrices, that is, greatest common divisors of minors of fixed order, we introduce the notion of type of a homomorphism $f$ of complex tori, which is similar to the type of a polarization. We show that the type is invariant under composition with isomorphisms, and that it completely describes the kernel of $f$ as a group. More precisely, the quotient of the kernel by its connected component containing 0 is a product of cyclic groups whose orders are determined by the type of $f$. Since the type can be computed from any rational representation of $f$, this gives an effective way to determine the kernel of a homomorphism. As a consequence, we compute the classic invariants degree and exponent of $f$, when $f$ has finite kernel. When $f$ is an isogeny, we also compute the type of its inverse isogeny from that of $f$. Finally, we compare the type of a polarization to the type of its associated isogeny.

Let $R(A,X)$ denote the largest ideal of a primitive axial algebra $(A,X)$ that contains no axis from the specified generating set $X$, and let $J(A)$ be the intersection of the maximal ideals of $A$. Mamontov, Shpectorov, and Zhelyabin asked whether $R(A,X)=J(A)$ always holds. We give a negative answer. Over every field of characteristic different from $2$, the two-dimensional commutative algebra with basis $a,b$ and multiplication $a^2=a$, $ab=2b$, and $b^2=b$ is a primitive axial algebra for an explicit fusion law and the generating set $X={a,b}$. Its complete ideal lattice is $0<\mathbb{F}b<A$, whence $R(A,X)=0$ and $J(A)=\mathbb{F}b$; in particular, the primitive axis $b$ lies in $J(A)$. Over $\mathbb{C}$, this axial presentation is equivalent to the previously classified $D(-1),{e_2,a_6}$ presentation with fusion law $F_{D3}$. Thus the algebra and axial structure are known; the new point is the computation of its Jacobson radical and the resulting counterexample to the radical-equality question.

We investigate finite non-degenerate set-theoretic solutions to the Yang–Baxter equation and skew braces using a probabilistic approach. We introduce four probabilities that measure how far a solution is from being a flip, but in different ways. Our main results state that for solutions arising from skew braces, these probabilities exhibit a rigid behaviour — apart from a finite list of exceptional values (which we show to occur by means of explicit examples), they admit an upper bound that is slightly above $\frac{1}{2}$. In the skew brace setting, these probabilities measure how far the underlying skew brace is from being a trivial brace (in two different ways: one via the annihilator and the other via the socle), a trivial skew brace, and an almost trivial skew brace. We also introduce a probability that is related to the indecomposable components of a solution, and a probability that measures how close an arbitrary bijective non-degenerate map is to being a solution. In contrast to our main results, these two probabilities do not exhibit a discrete behaviour near $1$ — they can be made arbitrarily close to $1$.

For every base field, we construct a finitely generated parafree augmented associative algebra with countably infinite-dimensional second homology. This disproves the analogue of the Parafree Conjecture for associative algebras and answers a question of Ivanov and Lopatkin. Our example is the monoid algebra of a finitely generated but not finitely presented submonoid of a free monoid.

Let $N\geq 3$. The class of $N$-Koszul AS regular algebras, or more generally, that of $N$-Koszul AS Gorenstein algebras, has attracted much attention from algebraists. Nevertheless, there have been no known examples of $N$-Koszul AS regular algebras of finite global dimension other than the ones of global dimension $3$. A recent work by Kabbaj showed that, such an $N$-Koszul algebra $A$ of finite global dimension has to have a large global dimension and that $N$ has to be prime, under the assumptions that $(1)$ $A$ has a Hilbert series of weighted polynomial rings and that $(2)$ the trivial $A$-module $\mathbb{K}$ has a finite free resolution. All AS regular algebras satisfy the latter assumption and are expected to do the former as well. In this paper, we prove that such an $N$-Koszul algebra $A$ must be one of the known $3$-Koszul AS regular algebras of global dimension $3$ if the order of the pole of its Hilbert series $h_A(t)$ at $t=1$ is greater than $\frac{21d+1}{22}$, where $d$ is the global dimension of $A$. As a corollary, we prove that any $N$-Koszul AS regular algebra $A$ must be one of the known $3$-Koszul AS regular algebras of global dimension $3$ if $A$ has a Hilbert series of weighted polynomial rings and if the GK dimension of $A$ coincides with the global dimension of $A$, both of which have been conjectured to hold for any AS regular algebras.

Let $k$ be a commutative ring, $H$ a bialgebra over $k$, $A$ an $H$-algebra and $\mathcal C$ an $(H,A)$-coring, i.e a left $(H,A)$-bimodule which is an $A$-coring in a compatible way. Left $(H,A)$-bimodules and their deformations play a fundamental role in the study of the differential geometry of noncommutative manifolds. Let $\Lambda$ be a right $(H,{\mathcal C})$-comodule. The $k$-module $_HEnd^{\mathcal C}(\Lambda)$ of right $(H,{\mathcal C})$-colinear maps from $\Lambda$ to $\Lambda$ is a ring. Let us assume that $\mathcal C$ is flat as a left $A$-module. If $\Lambda$ is finitely generated (finitely presented) as a right $(H,{\mathcal C})$-comodule, we give necessary and sufficient conditions for projectivity and flatness of a module over $_HEnd^{\mathcal C}(\Lambda)$. If $\mathcal C$ contains a fixed $H$-grouplike element, we can replace $\Lambda$ with $A$.

Let $A$ be a ring, let ${\bf T}$ be a finite-term big tilting complex over $A$, and choose a bounded projective representative $\cpx{P}$ of ${\bf T}$. Put $B=\End_{\D{A}}(\cpx{P})$. We prove that the kernel of the associated derived tensor functor ${\bf R}:\D{B}\to\D{A}$ is equivalent, through the derived inclusion, to $\D{\mathscr E}$, where $\mathscr E=\Ker({\bf R})\cap B\Modcat$ carries the inherited exact structure. Consequently, $\D{B}$ admits a recollement by $\D{\mathscr E}$ and $\D{A}$. The proof combines a dg double-centralizer argument with a functorial realisation of the kernel on unbounded complexes. The category $\mathscr E$ is definable and $d$-symmetric whenever $d$ is at least the projective amplitude of the underlying perfect right $B$-complex. We also characterise when the kernel is induced by a homological ring epimorphism: this holds precisely when $\mathscr E$ is an abelian subcategory, or equivalently when the standard $t$-structure restricts to the kernel. In amplitude at most one, this recovers the universal-localisation description for two-term big tilting complexes. Over semiperfect endomorphism rings, the amplitude is detected by reduction modulo the Jacobson radical. Finally, we construct non-compact examples of arbitrarily large minimal projective amplitude. One family yields algebraic Calkin quotients; another, over connected rings constructed from singular local hypersurfaces, has nonzero, non-abelian exact kernels and cannot be obtained from modules by derived equivalences between ordinary rings.

In this paper, we prove respectively that the disuccessor operations on the associative operad $\as$, the Lie operad $\lie$, and the pre-Lie operad $\prelie$ preserve Gröbner-Shirshov bases. This structural preservation enables the transfer of known bases to more complex operads. As a consequence, we introduce new methods for constructing Gröbner-Shirshov bases for the $\dend$ and $\prelie$ operads, and explicitly construct a Gröbner-Shirshov basis for the free L-dendriform algebra. This provides a conceptual and computationally efficient resolution of Madariaga's problem and offers new insights into the combinatorial structure of L-dendriform algebras.

Let $R$ be a ring. We organize the abstract elementary classes whose underlying class is the class of all $R$-modules and whose strong submodel relation lies between the submodule and direct summand relations into a lattice $\mathscr{L}_{R}$, ordered by reverse inclusion. We establish the basic lattice-theoretic properties of $\mathscr{L}_{R}$ and investigate its two natural sublattices, below and above purity. Below purity, we isolate relations defined by first-order pp-formulas for which amalgamation, tameness, and stability hold. Above purity, we introduce relations defined by infinitary pp-formulas and prove a broad stability result. Specializing to abelian groups, we show that the lattice $\mathscr{L}_{\mathbb{Z}}$ has the following properties: it has a strong submodel relation that is not positive syntactic, it contains an uncountable antichain and a strictly increasing proper-class-sized chain, and it has a broad region above purity where amalgamation fails.

We present a new impossibility region for the $5\times5$ symmetric nonnegative inverse eigenvalue problem. The region lies in the low-trace regime and, to the best of our knowledge, has not been identified previously. The proof uses a suitable diagonal shift to transform the problem to a critical high-trace boundary and then reduces a complementary commuting matrix to a weighted five-cycle. The characteristic polynomial and spectral identities of this five-cycle provide the main tools for deriving the resulting contradiction.

The titular object of this paper is an analogue of the Mandelbrot set over the 3-dimensional real algebra whose multiplication is the bilinear extension of the rock-paper-scissors operation. In order to study the dynamics of the mappings $x\mapsto x^2+c$ in this setting, a notion of holomorphy for functions on general finite-dimensional real algebras is introduced. Under a mild assumption it is shown that for such algebras holomorphy is always equivalent to solving a finite system of linear first-order PDEs generalizing the Cauchy-Riemann equations. Code is provided for generating animations of these fractals and for exploring them in a video game format.

Let $\mathcal V\subseteq M_6(\mathbb C)$ be a $17$-dimensional linear subspace such that $ \operatorname{rank}[S,T]\leq2 \quad(S,T\in\mathcal V). $ We prove that $\mathcal V$, or its transpose, is conjugate to the algebra $ \left\{ \begin{pmatrix} A&B&C\\ 0&\lambda I_2&D\\ 0&0&\lambda I_2 \end{pmatrix}: A,B,C,D\in M_2(\mathbb C),\ \lambda\in\mathbb C \right\}. $ Consequently, the corresponding closed algebraic locus in $\operatorname{Gr}(17,M_6(\mathbb C))$ is the disjoint union of two nonsingular irreducible components, each isomorphic to $\operatorname{Fl}(2,4;6)$. We also prove that the Zariski tangent space at $\mathcal A$ of the corresponding closed algebraic locus is equal to the tangent space to the conjugacy orbit of $\mathcal A$.

We say that a matrix over a finite field $\mathbb{F}_q$ is positive definite if it is symmetric and each of its leading principal minors is a nonzero square in $\mathbb{F}_q$. In previous work of the authors [J. Algebra, 2025], the entrywise positivity preservers on $M_n(\mathbb{F}_q)$ were classified in every case except when $n=2$, $q\equiv1\pmod4$, and $q$ is not a square. We settle this remaining case, thereby completing the classification of entrywise positivity preservers over every finite field and in every dimension $n\ge2$. Our proof is based on a novel idempotent reduction that not only resolves the remaining case but also yields a self-contained proof of the complete classification, while avoiding several technical results used in the earlier arguments. As a further application of the same reduction, we classify the entrywise preservers of strongly nonsingular matrices, i.e., matrices whose leading principal minors are all nonzero. We also prove a more general theorem in odd characteristic: for every prescribed sign pattern of nonzero leading principal minors of matrices of a fixed dimension $n\ge2$, the entrywise preservers are precisely the positive scalar multiples of field automorphisms. Thus, in odd characteristic, preserving any nonzero leading-principal-minor sign pattern surprisingly forces the preservation of every such sign pattern.

We investigate the structure of central extensions for algebras in a congruence modular variety. We use a multisorted algebraic object called a clonoid to understand the term clone of such a central extension. We develop the difference clonoid of such a central extension and use it to show that the number of $2$-step nilpotent algebras on a fixed finite set is finite if and only if the set is of squarefree order. The subpower membership problem for a finite algebraic structure $\mathbb{A}$ is the problem of deciding on input $a_1,\dots,a_k, b \in A^n$, whether $b$ is in the subalgebra of $\mathbb{A}^n$ generated by $a_1, \dots, a_k$. We show that for a large class of nilpotent Mal'cev algebras the subpower membership problem is solvable in polynomial time, in particular for $2$-step nilpotent Mal'cev algebras of squarefree order.

In 1996, Baranov and Kleshchev classified maximal ideals in the finitary symmetric group algebra over fields of characteristic not 2. The corresponding question in characteristic~2 has remained open since. In the current paper we solve the problem by establishing part of a conjectural connection between prime ideals in this group algebra and the recently defined higher Verlinde categories. To achieve this we investigate thick tensor ideals of tilting modules of the general linear group in characteristic 2.

For a ring $R$, let $\mathcal{GP}(R)$, $\mathcal{GF}(R)$, and $\mathcal{PGF}(R)$ denote the classes of Gorenstein projective, Gorenstein flat, and projectively coresolved Gorenstein flat left $R$-modules, respectively. We answer negatively the question whether $\mathcal{GP}(R)=\mathcal{PGF}(R)$ for every ring. More precisely, over every field $k$ we construct a left and right coherent central $k$-algebra $T$ and a strongly Gorenstein projective left $T$-module which is not Gorenstein flat; hence $\mathcal{PGF}(T)\subsetneq\mathcal{GP}(T)$.

We classify the entrywise functions that preserve positive semidefiniteness on discrete Green matrices \(G(p,q)=(p_{\min(i,j)}q_{\max(i,j)})\) with positive parameters, without requiring the resulting matrix to retain Green structure. For matrices of all orders, the preservers are the zero function and the functions \(f(t)=\int_{[0,\infty)}t^\alpha\,d\mu(\alpha)\), where \(\mu\) is a nonzero finite positive measure and the integral is finite for every \(t>0\). Requiring the resulting matrix to be totally nonnegative reduces the nonzero preservers to \(f(t)=ct^\alpha\), where \(c>0\) and \(\alpha\ge0\). These power functions also preserve positive semidefinite Green structure, while strict Green structure is preserved precisely when \(\alpha>0\). No regularity assumption is needed for these classifications. We also characterize continuously differentiable functions that are entrywise Loewner monotone on every fixed-\(q\) Green family: this holds precisely when \(f'\) is a positive mixture of nonnegative real powers, with the zero measure allowed.

We prove the nonexistence of eight circulant weighing matrices from the remaining table of orders at most $200$ and weights at most $100$. The proofs combine contraction, character evaluation on the kernel of a contraction, multiplier methods, and exact finite computations. For $CW(105,36)$, the contracted matrix is unique up to equivalence. Applying a nonprincipal character of the $C_3$ kernel gives an element over the Eisenstein integers; reduction modulo $1-\omega$ gives a word in a ternary cyclic code of length $35$, and exact enumeration rules out every required Eisenstein-unit lift. For $CW(140,36)$, the real-valued character $Y\mapsto-1$ of the $C_4$ kernel is incompatible with the same contracted class. For weight $64$, the faithful character $Y\mapsto i$ of a $C_4$ kernel first gives an element of $\mathbb{Z}[i][C_m]$; a generalized multiplier then forces constancy on multiplication-by-$2$ orbits, and exact correlation calculations eliminate orders $140$, $180$, and $196$. The three weight-$49$ cases are settled by the ordinary prime-power multiplier, with contraction where needed. Consequently none of $CW(105,36)$, $CW(140,36)$, $CW(116,49)$, $CW(120,49)$, $CW(192,49)$, $CW(140,64)$, $CW(180,64)$, and $CW(196,64)$ exists.

We study subcategories of the module category of a finite-dimensional algebra that are closed under quotients or submodules. We propose the quotient–submodule equidistribution conjecture: over a representation-finite algebra, the number of quotient-closed subcategories of size $i$ is equal to that of submodule-closed subcategories of size $i$ for every $i$, where size is the number of indecomposable modules in the subcategory. We prove the following cases of the conjecture: (1) the five smallest and five largest values of $i$, hence all algebras with at most nine indecomposable modules; (2) Nakayama algebras; (3) algebras with radical square zero; and (4) representation-directed algebras. In the last case, we classify quotient-closed subcategories by a lower Bruhat interval in a Coxeter group constructed from the Auslander–Reiten quiver, extending the classification of Oppermann–Reiten–Thomas for Dynkin quivers. We also prove that quotient closure defines a finitary convex geometry and classify the functorially finite quotient-closed subcategories by Gen-minimal modules.

For Rota-Baxter Lie algebras, a homotopy cooperad is exhibited, whose cobar construction is shown to be the minimal model of the operad of Rota-Baxter Lie algebras by using algebraic Morse theory. The deformation complex of Rota-Baxter Lie algebras as well as the $L_\infty$-algebra structure on this complex are deduced from the minimal model and the notion of homotopy Rota-Baxter Lie algebras is given as a consequence.

For a ring $R$, let $\mathcal{GP}(R)$, $\mathcal{GF}(R)$, and $\mathcal{PGF}(R)$ denote the classes of Gorenstein projective, Gorenstein flat, and projectively coresolved Gorenstein flat left $R$-modules, respectively. We isolate the local ultrafilter hypothesis $\textsf{LUH}$: the existence of a strongly compact cardinal implies $\textsf{LUH}$, while $\textsf{LUH}$ implies the existence of a measurable cardinal. Assuming $\textsf{LUH}$, we construct a left and right coherent ring $R$ and a strongly Gorenstein projective left $R$-module $G$ which is not Gorenstein flat; hence $\mathcal{PGF}(R)\subsetneq\mathcal{GP}(R)$.