Papers

The aim of the paper is to study Euclidean and l-finite Euclidean modules over a ring, submodules of their finite direct sums, reduced and strong reduced echelon forms of matrices with coefficients in Euclidean modules and bimodules. The concept of Euclidean module is a module-theoretic analogue of the concept of Euclidean ring. The class of (not necessarily commutative) Euclidean rings is a nice but `very small' class of rings that have arithmetic-like properties. On the contrary, surprisingly, the class of Euclidean modules is a `large class' of modules. The Euclidean modules exist for many constructions of rings that are defined over arbitrary rings. For example, for every skew polynomial ring $A=D[x;\s, \d]$ with coefficients in an arbitrary ring $D$, the generalized Weyl algebra $D[x,y;\s, a]$, the Weyl algebra $A_1=K[x][\der; \frac{d}{dx}]$, $U(\sl2)$ and many classical algebras, there exist Euclidean modules.

Schreyer's algorithm is usually the fastest way to determine a (typically non-minimal) free resolution of a finitely presented module over a polynomial ring. We adapt a refined version of Schreyer's algorithm to compute free resolutions over the exterior algebra, relying on relative Groebner bases. We illustrate the use of our algorithm for the computation of cohomology of coherent sheaves over projective space, which by the BGG correspondence can be computed via free resolutions over the exterior algebra. Schreyer's method relies on a tree traversal and thus has the potential for parallel computations. We report on ongoing work on a massively parallel implementation, observing that Gnawali's parallel approach over polynomial rings can be carried over to our setting.

We introduce and study the algebraic Dirac operator for the Cartan motion group associated with a real reductive Lie group. We establish its fundamental properties, including a square formula involving a natural Casimir-type element, define the corresponding notion of Dirac cohomology, and prove a strong form of Vogan's conjecture: a module in the Dirac series is completely determined by its Dirac cohomology. We explicitly determine the Dirac series of a Cartan motion group. We explain how the Dirac theory of a Cartan motion group can be obtained as the zero fiber of the recently introduced Dirac theory for the algebraic deformation family of the corresponding real reductive group.

The Wagner-Preston theorem states that every inverse semigroup can be embedded into an inverse semigroup of partial bijections on some set. However, this embedding does not respect lattice operations in the natural ordering. In this paper, we show that an abstract pseudogroup can be faithfully represented by partial bijections on a set if and only if its frame of idempotents is spatial. More generally, we show that any abstract pseudogroup can be faithfully represented by partial homeomorphisms on a locale.

Let $\operatorname{UT}_n$ denote the group of $n\times n$ real upper triangular matrices with positive diagonal entries. We derive a closed-form analytic expression for the Karcher mean on $\operatorname{UT}_2$ and obtain a sharp convergence threshold for the projection mean iteration with fixed initial diagonal, together with a sufficient convergence condition for arbitrary initial points. In addition, we obtain explicit formulas for the log-Euclidean mean and establish a Lie–Trotter type formula for the Karcher mean on $\operatorname{UT}_2$. For $\operatorname{UT}_n$ we derive the diagonal and first superdiagonal entries of every solution of the Karcher equation and transfer nonconvergence examples from $\operatorname{UT}_2$.

Let $d_1,d_2,\ldots, d_r\ge 2$ be $m$-th power free positive integers. In this paper, we provide explicit representations of all irreducible $\mathbb{Z}_+$-modules over $\mathbb{Z}[\sqrt[m]{d_1},\sqrt[m]{d_2},\ldots,\sqrt[m]{d_r}]$, where $\mathbb{Z}_+$ denotes the semiring of non-negative integers.

We investigate the problem of which multisets of integer entries yield integer eigenvalues in every square matrix arrangement? Combining module reduction with a dyadic criterion for complete splitting of cubic polynomials, we first show that if the multiset has at least one zero entry, the only possibilities have either at most one nonzero entry or exactly two whose product is a perfect square. We then apply block embedding to extend it to higher dimensions, by obtaining a linear threshold on the number of zeros. For any multiset close to a nonzero constant, we use rank reduction to derive an exact criterion for one exceptional entry, thus yielding nonconstant examples in infinitely many dimensions, and to also derive an exact reduction for two distinct exceptional entries in dimension three. The general fully nonzero three-dimensional case remains open.

We explicitly investigate Gorenstein projective, injective and flat modules over the $n$-trivial extension $R\ltimes_{n}M$ of a ring $R$ by an $R$-bimodule $M$. Assume that $fd(M^{\otimes_{R}i}_{R})<\infty$ and $pd(_{R}M^{\otimes_{R}i})<\infty$ for any $1\leq i\leq n$, $fd(\textbf{Z}(R)_{R\ltimes_{n}M})<\infty$ and $pd(_{R\ltimes_{n}M}\textbf{Z}(R))<\infty$. It is proven that a left $R\ltimes_{n}M$-module $(X,f)$ is Gorenstein projective if and only if the sequence $M^{\otimes_{R}n+1}\otimes_{R} X\stackrel{(M\otimes f) \cdots(M^{\otimes_{R}n}\otimes f)}\longrightarrow M\otimes_{R} X\stackrel{f}\longrightarrow X$ is exact and coker$(f)$ is a Gorenstein projective left $R$-module. As a consequence, we characterize Gorenstein projective, injective and flat modules over tensor rings.

We exhibit two vector fields on $S^2$, which generate the entire Lie algebra of algebraic vector fields. Compositions of the flows associated with these two vector fields enable us to approximate every diffeomorphism of $S^2$ isotopic to the identity in the $\mathrm{C}^r$-sense for any $r\geq 0$.

Let $R$ be a ring and $M$ a left $R$-module. We prove that $M$ is an $\omega_1$-indexed inverse limit of injective modules with split epimorphic connecting maps, and $M$ is also an $\omega$-indexed inverse limit of a stationary tower defined by a split monic endomorphism of one injective module. These constructions answer two questions of Bergman. We then study the class of direct summands of $\omega$-indexed inverse limit of injective modules with epimorphic connecting maps. Over a commutative Noetherian ring $R$, they are precisely the Enochs cotorsion modules if and only if $\operatorname{gldim}R\leq1$ and $\operatorname{Spec}R$ is countable.

Let $R$ be a commutative Artinian ring. We consider two graphs associated to $R$, namely the compressed zero-divisor graph $\Gamma_E(R)$ and the associate class graph $\Gamma_A(R)$. Partitioning the vertex set of a zero-divisor graph into its core and its boundary, we count the core vertices that dominate the core. This count is a graph invariant, and we estimate it for $\Gamma(R)$, $\Gamma_A(R)$ and $\Gamma_E(R)$. We prove that the count for $\Gamma_A(R)$ is bounded below by the count for $\Gamma_E(R)$, and that the lower bound is attained precisely when $R$ is Gorenstein. As a consequence we obtain that $R$ is Gorenstein if and only if $\Gamma_A(R)\cong\Gamma_E(R)$ as graphs, the isomorphism being an arbitrary one and not merely the natural compression map. Using the same counting technique we then answer, for Artinian rings, a question of Anderson and LaGrange by showing that $\Gamma(R)\cong\Gamma_E(R)$ if and only if $R\cong \mathbb Z_2^{\,n}$ for some $n\ge2$, or $R\cong\mathbb Z_4$, or $R\cong\mathbb Z_2[x]/(x^2)$.

We extend recent work on analogues of Grün's Lemma from skew-braces to all semiassociative Mal'cev algebras which is a broad class of algebras encompassing all expansions of groups. The direct analogue of Grün's Lemma is known to fail for skew-braces; thus, the more general result is a characterization for perfect algebras of when the upper-central series stabilizes at the second-center. The proof is an application of the description of generating sets for commutators in Mal'cev algebras.

We prove Keller's conjecture for every split finite-dimensional algebra over an arbitrary field.

Matrix Hilbert series retain local elementary-divisor data that their determinants discard. For locally finite positively graded elementary algebras whose vertex simples are of type $FP_\infty$ and whose matrix Hilbert series is rational, we construct from possibly infinite minimal resolutions a rational Euler matrix and its signed local Smith profile at $x=1$. The profile refines pole-order growth for perfect modules and is invariant under shift-compatible graded Morita and perfect-derived equivalences. Every finite integer profile occurs, while a wholly negative profile forces exponential corner growth. This extends the nonnegative local Smith framework for twisted Calabi–Yau algebras to a setting in which the Euler matrix may itself have poles.

We settle in the negative the longstanding question whether every Bézout domain is an elementary divisor domain by constructing a Bézout domain over which an explicit $2\times 2$ matrix has no Smith normal form. The obstruction is topological and is detected by the Möbius line bundle.

We establish strong and explicit forms of a one-sided Nullstellensatz for polynomial rings D[x1, . . . , xn] over arbitrary division rings D with central indeterminates. To this end, we employ the theory of integral dependence over left ideals and the notion of G-left ideals, which plays a role analogous to that of G-ideals in the classical theory. We prove that the Jacobson radical of any left ideal of D[x1, . . . , xn] coincides with the intersection of all G-left ideals containing it. As a consequence, we obtain an explicit description of Jacobson radicals of left ideals of polynomial rings over division rings in terms of integral dependence.

We study the weighted Karcher mean on the Lie group $\operatorname{UT}_n$ of upper triangular matrices with positive diagonal entries. Using a divided-difference description of analytic functions on upper triangular matrices and the Parlett recurrence, we prove the existence, uniqueness, and analyticity of the weighted Karcher mean and derive recursive formulas for its computation. The theory yields explicit expressions in low dimensions and formulas for weighted geometric means on $\operatorname{UT}_n$. More generally, the results reveal a close relationship between divided differences, analytic matrix functions, and nonlinear matrix means on triangular Lie groups.

We introduce the notion of a quasi-cellular category, which is a variant of cellular algebras and cellular categories. Many diagram categories, such as the Brauer category and the partition category, are quasi-cellular. We study the Brauer category $B$ over a commutative ring $\mathbb k$ with parameter $\delta$ through its quasi-cellular structure. We construct a linear functor $L:B\to mB$ to the matrix Brauer category $mB$, which has the same objects and hom-spaces as $B$ but a matrix-like composition. For a non-singular parameter $\delta$, we show that $L$ is an isomorphism of linear categories, and that $B$ is Morita equivalent to its subcategory $\mathbb S$ spanned by permutations, whose endomorphism algebras are the group algebras of symmetric groups. These results generalize those of Brown and König and Xi for the Brauer algebras.

Keller's strong form of the homological conjectures asserts: any finite dimensional algebra cogenerates its unbounded derived category of modules. Here we record an example of a (coconnective) dg algebra with finite dimensional cohomology, which does not cogenerate its module category, along with some related phenomena: a smooth dg category whose dualizing bimodule fails to be nondegenerate, and a nontrivial fully faithful left Calabi-Yau morphism. All examples and most proofs were produced by ChatGPT. In an appendix we explain the reason we were looking for such examples: their existence would follow from the existence of Weinstein symplectic manifolds which failed to satisfy Arnol'd's chord conjecture.

Let \(H\) be a multiplicative group, \(R\) a commutative \(H\)-graded ring, and \(M\) a nonzero finitely generated gr-semisimple \(H\)-graded \(R\)-module. We assume that \(M\) admits a fixed decomposition into powers of gr-simple modules with pairwise distinct annihilators, each repeated block carrying the componentwise grading. For a nonzero homogeneous element \(x\), we compute the colon ideal \((Rx:M)\) and show that it is determined by the multiplicity-one blocks on which \(x\) has a nonzero component. This leads to the criterion \( x\leftrightarrow y \text{ if and only if } \sigma(x)\cap\sigma(y)=\varnothing, \) where \(x\leftrightarrow y\) means that \(x\) and \(y\) are adjacent in \(\Gamma_H(M)\). Thus adjacency is governed by disjointness of multiplicity-one obstruction supports. The corresponding obstruction classes form a weighted disjointness structure: higher multiplicities affect the sizes of the classes, but not the adjacency relation between their supports. We use this description to study graph invariants, reconstruction, and Kneser-type layers, and, in the finite case, to obtain formulas for the obstruction-class sizes and a reduction of the adjacency spectrum.

Let $T$ be a nonzero real symmetric third-order tensor. Denote by $r(T)$ its CP rank, by $s(T)$ its symmetric rank, by $d(T)$ its essential dimension, and put $e(T)=r(T)-d(T)$. We establish two complementary upper bounds for the Comon rank gap $$\Delta(T):= s(T)-r(T)\leq\min\{C_{ec}(e(T)),\ C_{dc}(d(T))\}.$$ The excess-controlled function $C_{ec}(e)$ is zero for $0\leq e\leq3$ and equals $9e-3$ for $e\geq4$. In particular, the equality $C_{ec}(e(T))=0$ for $0\leq e\leq3$ extends Friedland's flattening-rank criterion for real symmetric third-order tensors by two additional rank layers. The dimension-controlled function $C_{dc}(d)$ is zero for $d=0$, equals $(d^2-4d+10)/3$ for $1\leq d\leq18$, and equals $d^2/4+d/4+3/2$ for $d\geq19$. The excess estimate follows from the alternating equations and decomposable-bivector support geometry; a low-wedge descent gives the vanishing range $e\leq3$. The dimension estimate combines a transverse polar-completion bound with a shifted estimate for the maximum real Waring rank. Thus the two bounds measure different features of the same tensor and remain effective in complementary regimes.

We give a direct, self-contained construction of the three sporadic Fischer groups $\mathrm{Fi}_{24}'$, $\mathrm{Fi}_{23}$, and $\mathrm{Fi}_{22}$ from the $783$-dimensional Conway-Parker algebra. We prove that its distinguished roots define involutory algebra automorphisms whose projective actions generate the full Fischer $3$-transposition group $\mathrm{Fi}_{24}$. Its commutator subgroup gives $\mathrm{Fi}_{24}'$, while $\mathrm{Fi}_{23}$ and $\mathrm{Fi}_{22}$ arise as centralizer quotients associated with one and two commuting transpositions. The root and frame geometry determines the group orders and leads to elementary proofs of simplicity, as well as natural rank-three actions and nonsplit central extensions. The construction uses standard facts about the Golay code, Parker's loop, and $M_{24}$. It does not use the Monster or previously known existence or order results for the Fischer groups. Fischer's classification and later recognition theorems are used only for the final identification.

We classify the Classifying Grothendieck rings of pivotal fusion categories of rank four over the complex field. There are exactly fifteen, each admitting a unitary categorification. Central induction and Frobenius-Schur indicators give a uniform Frobenius-Perron dimension bound of 3600. The finite classification combines new arithmetic and twist obstructions with an exhaustive census and exact exclusion certificates.

Strict unimodality of Ooms spectra fails for Frobenius parabolic Lie algebras beyond the maximal-parabolic class. We construct explicit Frobenius parabolics whose principal multiplicity sequences contain arbitrarily long outward rises: for every $L\ge1$ there are $L$ consecutive negative first differences, each of quadratic size in $L$. In a simple one-parameter subfamily the transition to nonunimodality occurs in $\slie_{136}$, where two adjacent multiplicities are $541<542$. We also give a representation-theoretic criterion for unimodality. If $F$ is a Frobenius functional with semisimple integral principal element $H$ and $\g=\bigoplus_j\g_j$, the Kirillov form makes the centered grading symplectic and its character has the unique expansion \[ \Theta_\g(x)=\sum_{m\ge1}(\dim\g_m-\dim\g_{m+1})\,\chi_{2m-1}(x) \] in irreducible $SL_2$ characters. Thus Ooms unimodality is equivalent to effectivity of this virtual $SL_2$ character, or equivalently to extension of the centered principal cocharacter to a symplectic $SL_2$ action. For Frobenius seaweeds the unbroken-spectrum theorem says that, after centering, every odd weight in the support occurs; strict unimodality asks for the stronger positivity of every irreducible coefficient. A companion paper proves that stronger property for every Frobenius maximal parabolic. The families constructed here show that beyond that class there can be arbitrarily long blocks of negative irreducible coefficients, with unbounded total negative mass.

Chu proved that a symplectic structure $\omega$ on an even-dimensional Lie algebra induces a left-symmetric structure which is defined by $\omega(x\scriptstyle{\Delta}_{\omega} y,z)=-\omega(y,[x,z])$. El Bourkadi and Mansouri proved that a cosymplectic structure on an odd-dimensional Lie algebra $\mathfrak{g}$ induces the left-symmetric structure, which is defined by using a linear isomorphism from $\mathfrak{g}$ to $\mathfrak{g}^*$ associated with the cosymplectic structure. In this paper, for a non-degenerate bilinear form $\phi$ on a Lie algebra $\mathfrak{g}$, we give a necessary and sufficient condition for the product $\scriptstyle{\Delta}_{\phi}$ on $\mathfrak{g}$ defined by $\phi(x\scriptstyle{\Delta}_{\phi} y,z)=-\phi(y,[x,z])$ to be a left-symmetric structure. We also prove that the left-symmetric structure $\scriptstyle{\Delta}_{\phi}$ is complete if and only if the Lie algebra $\mathfrak{g}$ is unimodular. Moreover, we formulate the notion of double extension for non-degenerate bilinear forms and prove that a certain class of non-degenerate bilinear forms is obtained by double extension.

We prove universal positivity of rank-two quantum greedy elements when $b\mid c$ or $c\mid b$, resolving a conjecture of Lee, Li,Rupel and Zelevinsky.

The authors consider several variants of the notion of nilpotency in pre-Lie algebras and investigate the properties of nilpotent ideals.

We prove that every finite-dimensional commutative algebra can be embedded as an ideal of a finite-dimensional evolution algebra. We then determine the smallest possible dimension of an evolution algebra admitting such an embedding and study some structural properties preserved by the construction. Over fields of characteristic different from two, we relate this minimal dimension to simultaneous Waring decompositions of the quadratic map $x\mapsto x^2$. In particular, for unital commutative algebras, the smallest possible dimension of an evolution algebra containing the given algebra coincides with the Waring rank of its squaring map. As an application, we determine this minimal dimension for the truncated polynomial algebras $\mathbb{F}[t]/(t^n)$. Finally, we study embeddings for which additional structure is preserved, including automorphisms and multiplicative bases.

Let $B$ be a finite-dimensional algebra, $M\in\modu B$, and $A=B[M]$ the one-point extension. We give a stable two-step extension criterion for the new simple $A$-module to have derived delooping level at most one. For a concrete algebra $B$ with a self-injective Nakayama quotient $R$, we determine the derived and classical delooping levels of $B[M]$ for every nonzero $M\in\modu R$. In particular, we obtain an explicit algebra $A$ satisfying \[ \Findim(A^{\mathrm{op}})=1<2=\ddell A=\dell A, \] which gives a negative answer to a question of Guo and Igusa.

We show that the constraint satisfaction problem of any finite conservative structure that can not pp-construct st-Con (whose CSP is NL-complete) or 3Lin2 (whose CSP is Mod2L-complete) is solved by a symmetric linear Datalog program and therefore in L.