Papers

Motivated by the equivalent definitions of positive definite (resp. positive semidefinite) matrices over real and complex fields, we give four (resp. five) inequivalent definitions for these matrices over finite fields. Our starting point is the recent definition due to Cooper–Hanna–Whitlatch (RMJ. Math., 2024) of positive elements in finite fields. We also use this definition to study totally positive matrices over finite fields. For all of these cases, we give explicit enumeration formulae or give bounds. Most of our formulas are new, but we summarize results from the existing literature for completeness. For positive semidefinite matrices of type 5 and totally positive matrices, we give structural formulas using the rationality of the Weil zeta function, i.e. Dwork's theorem, and conjecture a quasipolynomial-type formula.

In this note, we give a complete characterization of square matrices of size $n$ over a commutative field which can be expressed as a product of a nilpotent matrix and an involutive matrix and matrices which can be expressed as a product of a nilpotent matrix and an idempotent matrix, in terms of invariant factors.

We investigate the subspaces obtained by images of multilinear graded polynomials evaluated on the matrix algebra endowed with the canonical $C_n$-grading, where $C_n$ denotes the cyclic group of order $n$. Moreover, this graded algebra admits a natural action of $C_n$ arising from a fine refinement of the grading. We prove that every $C_n$-submodule of the trivial component of the grading is the image of some multilinear polynomial in graded variables. In particular, we answer in the negative a recent conjecture posed by T.~de Castilho and L.~Centrone (2023).

A two-parameter hierarchy of classes of finitely generated modules over an artin algebra is described by using the $\mho$-quiver. As an application, it is shown that the stable category of reflexive modules is equivalent to two other categories via the mutually quasi-inverse equivalences induced by $\Omega$ and $\mho$.

Let $L$ be a finite-dimensional Lie algebra over a field $F$. The comaximal graph $\Gamma(L)$ has as vertices the proper nonzero subalgebras of $L$, two of them adjacent whenever they generate $L$; its structure was previously classified for Lie algebras of dimension at most 3 over finite fields. Here we extend that work in two directions. First, we obtain explicit formulas for the number of triangles $t(\Gamma(L))$ for every three-dimensional Lie algebra over $\F_q$. Second, we extend the classification to several four-dimensional families over $\mathbb{F}_q$, the abelian, Heisenberg, and filiform algebras, and $\mathfrak{gl}_2(\F_q)$. We also relate graph-theoretic properties of $\Gamma(L)$, such as completeness and the role of the Frattini subalgebra, to structural properties of $L$, including supersolvability. These results yield new combinatorial invariants for finite-dimensional Lie algebras over finite fields.

Let R be a commutative semiring, not necessarily with a multiplicative identity, and let A be an element of Mn(R). We prove that the map x to Ax on Rn is injective if and only if x to AT x is injective. Equivalently, the left- and right-cancellative elements of the multiplicative semigroup Mn(R) coincide. The proof splits formal determinant expansions into their even and odd halves; it uses no subtraction, additive cancellation, group completion, inverse, or multiplicative identity. As consequences we recover the stable-finiteness theorem for matrices over unital commutative semirings. We also prove that surjectivity is invariant under transpose. In fact, the existence of a surjective square matrix of positive size forces R to have a multiplicative identity.

Let $A$ be a module-finite algebra over a commutative Noetherian ring. We give a local criterion for the $n$-Gorenstein condition. For such algebras satisfying the Auslander condition, we prove that finite self-injective dimension is symmetric. Under a one-sided local finiteness hypothesis, we further characterize the Auslander condition by a fiberwise form of Iyama's grade bijection and obtain a criterion for the Auslander–Gorenstein property.

In arXiv:1405.4105 it was asked whether an affine Hopf algebra with finite Gelfand-Kirillov dimension is necessarily Noetherian. It is well-known that the converse is not true –take the group algebra of a polycyclic group which is not nilpotent-by-finite. However, it was conjectured in arXiv:2301.04428 that a pointed affine Noetherian Hopf algebra whose group of group-likes is nilpotent-by-finite necessarily has finite Gelfand-Kirillov dimension. In the present paper we conjecture that a post-Nichols algebra over a polycyclic-by-finite group is Noetherian if and only if it is affine and has finite Gelfand-Kirillov dimension. Partial results supporting this conjecture are presented; and their consequences for the preceding questions and conjectures is analyzed.

We develop an induction theory for relative BiHom-Hopf modules, the special BiHom-Doi–Hopf case associated with the datum $(H,A,H)$ in which the final copy of $H$ carries its regular right $H$-module coalgebra structure. Let $H$ be a monoidal BiHom-Hopf algebra, let $A$ be a right $H$-BiHom-comodule algebra, and let $B=A^{\operatorname{co}H}$. We show that the balanced tensor product defines an induction functor $-\otimes_BA$ left adjoint to the coinvariant functor. If $H$ has a fixed Haar integral, untwisting yields a Haar identity for the BiHom setting and a canonical projection onto coinvariants. These constructions define a hereditary torsion theory with radical $\kappa$ and a torsion-free reflector $Q(M)=M/\kappa(M)$. The modified induction functor $Q(-\otimes_BA)$ then gives an equivalence between right $B$-BiHom-modules and torsion-free relative BiHom-Hopf modules generated by their coinvariants. Equal structure maps recover the corresponding Hom result, while identity structure maps recover the classical relative Hopf-module setting.

A $\varphi$-algebra is either a superalgebra or an algebra with involution. In this paper, we study $\operatorname{T}^\varphi$-ideals associated with unital $\varphi$-algebras whose $\varphi$-codimension sequence exhibits cubic polynomial growth. As a consequence, we obtain a complete classification of all $\varphi$-codimension sequences of cubic growth for unital $\varphi$-algebras. Furthermore, we explicitly determine a minimal-degree multilinear generator for every $\operatorname{T}^\varphi$-ideal associated with unital $\varphi$-algebras whose $\varphi$-codimension growth is at most quadratic.

We combine results of Rickard and Shaul with methods of intrinsic homological algebra and the theory of purity to prove that the finiteness of the big finitistic dimension of an Artin algebra is an intrinsic property of the singularity category of its opposite algebra, via its category of modules. This `object-free' approach complements work of Dey–Šťovíček and arrives at the same conclusion: the finiteness of the finitistic dimension of Artin algebras is a singular invariant (of the opposite algebras). In fact, our characterisation can be used to prove that finite finitistic dimension descends along certain fully faithful functors between singularity categories. We include an appendix where we explain how our methods can be used to prove non-existence of bounded t-structures on singularity categories.

Let $R$ be a field and $A$ the endomorphism algebra of a finite direct sum of cyclic modules over a finite-dimensional commutative local principal ideal $R$-algebra. We construct a central quotient showing that the support $\tau$-tilting poset of $A$ is isomorphic to the poset of the symmetric group with the weak order. We show that the center of $A$ and degree-zero Hochschild homology, viewed as a module over the center, determine the truncated local algebra and the multiset of successive length gaps. For centralizer matrix algebras, the support $\tau$-tilting poset determines the multiset of distinct-exponent counts of the primary blocks. The corresponding algebra–module pair also recovers their local algebras and gap multisets. Combining this reconstruction with the known derived equivalence classification, we characterize derived equivalence by isomorphism of these algebra-module pairs. We apply the results to Morita reconstruction in the string and gentle classes.

A real symmetric matrix is called $d$-locally positive semidefinite if all of its $d \times d$ principal submatrices are positive semidefinite. We investigate the spectral geometry of $d$-locally positive semidefinite matrices. The set of vectors of eigenvalues of $d$-locally positive semidefinite matrices of size $n \times n$ is fully understood and known to be convex when $d \in \{1,n-1,n\}$ \cite{blekherman2022hyperbolic}. In the smallest remaining case $n=4, d=2$, non-convexity of the set of vectors of eigenvalues was proved in \cite{kozhasov2023eigenvalues}, but even in this case the full description was unknown. We provide a basic semialgebraic description of the set of vectors of eigenvalues for $n=4, d=2$ by establishing a Fischer-type inequality for $2$-locally positive semidefinite matrices of size $4 \times 4$ and prove non-convexity for $n \geq 4$ and $d \in \{2, n-2\}$. Non-convexity is established via solving certain non-smooth and non-convex min-max point configuration problems in the complex plane, which could be interesting in themselves. Similar problems were considered in \cite{nesterenko2024submatrices,sengupta2026submatrices} in the context of matrix decomposition and approximation.

This paper investigates the structural properties, operator representations, and algebraic duality of Hom-Novikov superalgebras and to introduce Hom-anti-pre-Novikov superalgebras. We establish a formal equivalence between Hom-Novikov superalgebras and admissible Hom-Novikov superalgebras via $q$-superalgebra extensions, proving that a Hom-pre-Lie superalgebra admits a Hom-Novikov structure if and only if its associated $2$-superalgebra is admissible. Introducing $q$-admissible pairs $(\mathcal P, \mathcal Q)$ on super-commutative Hom-associative superalgebras, we detail systematic constructions of Hom-Novikov superalgebras and their admissibility classes. Furthermore, we characterize anti-super-$\mathcal{O}$-operators on Hom-Lie superalgebras, showing that invertible operators induce compatible Hom-anti-pre-Lie structures and extend directly to admissible setups. Extending these frameworks to Poisson-type structures, we demonstrate that the sub-adjacent Hom-Lie superalgebra of a Hom-anti-pre-Lie Poisson superalgebra yields a transposed Hom-Poisson superalgebra. Finally, we establish the fundamental connection between Hom-anti-pre-Novikov superalgebras and invertible anti-super-$\mathcal{O}$-operators.

We study some structural properties of Lie affgebras as affine analogues of Lie algebras. We introduce the notion of an ideaf, the affine counterpart of an ideal, and establish characterizations of left and right ideafs. We further study centers, quotient structures, and product ideafs, proving, under suitable conditions, that the center of a Lie affgebra is an ideaf. We then develop the notion of affine nilpotency and establish connections between the nilpotency of Lie affgebras and that of their retracted Lie algebras. As an application, we prove an Engel-type theorem for Lie affgebras of the form $\mathfrak{a}(\mathfrak{g};\kappa=2\lambda,\lambda,s)$.

In this paper, we investigate the structure of locally generalized radical linear groups over certain non-commutative rings and algebras. The base rings and algebras we consider include division rings, left artinian rings, locally finite algebras over fields, and PI-algebras over fields. Among various results, we get the positive answers to the General Burnside Problem and to Baer's Conjecture for some particular cases of linear groups.

Let \[ A_{\min}(x)=\bigl(x_{\min(i,j)}\bigr)_{i,j=1}^n, \qquad A_{\max}(x)=\bigl(x_{\max(i,j)}\bigr)_{i,j=1}^n \] be the Min and Max matrices generated by a real sequence \(x=(x_1,\ldots,x_n)\). Using their classical cone parametrizations, we give exact characterizations of entrywise maps preserving positive semidefiniteness, total nonnegativity, Loewner order, and Loewner convexity. Our main results concern the Loewner structure.Without assuming continuity or differentiability, entrywise Loewner-order preservation is equivalent to \(f\) being nondecreasing and convex. On the Min and Max cones, requiring the Loewner-convexity inequality on arbitrary pairs is rigid and forces \(f\) to be affine. Under the standard convention of restricting the inequality to Loewner-comparable pairs, the condition automatically forces \(f\in C^1([0,\infty))\) and is equivalent to convexity of both \(f\) and \(f'\). The analogous statement holds for Loewner concavity. In each case, the condition is already detected in dimension two. We also show that \(f:[0,\infty)\to\mathbb R\) preserves positive semidefiniteness entrywise on all positive semidefinite Min or Max matrices if and only if \(f\) is nonnegative and nondecreasing; the same condition characterizes total-nonnegativity preservation. Finally, we determine the power-function ranges and Loewner-order automorphisms, characterize the entrywise preservers of the strict Min and Max classes, and relate these strict cones to inverse \(M\)-matrices and oscillatory matrices.

We study the class of congruences on $F$-inverse monoids that respect the unary operation $a\mapsto\mx{a}$, assigning to each element the maximum element of its $\sigma$-class, which we refer to as $F$-congruences. We characterize kernel normal systems and congruence pairs on $F$-inverse monoids which give rise to such congruences and establish a number of further related results including a classification of $F$-congruence-free $F$-inverse monoids.

We study stable equivalences of Brauer graph algebras. In particular, we prove that Brauer graph algebras are closed up to semisimple summands under stable equivalence of Morita type. As a consequence, we reprove the result of Antipov and Zvonareva that Brauer graph algebras are closed under derived equivalence. As a byproduct, we get a solution of the reconstruction problem posed by Rickard and Rouquier for algebras stably equivalent of Morita type to Brauer graph algebras.

By using an advanced model of AI, it is proved that there exists a commutative ring having a purely-prime ideal which is not purely-maximal. This gives a counterexample to \cite[Conjecture 5.8]{Tarizadeh}.

We introduce cleft extensions for Hopf algebroids. We prove the equivalence between cleft extensions, $\sigma$-twisted crossed products, and Hopf-Galois extensions with the normal basis property, thereby generalizing the theory of cleft extensions for Hopf algebroids developed by Böhm and Brzeziński, and fitting in with the general theory of Galois and biGalois extensions over Hopf algebroids developed by the authors. We investigate the Ehresmann Hopf algebroid associated with a cleft extension and show that it is isomorphic to a generalized version of the Connes-Moscovici Hopf algebroid. A special case of the Connes-Moscovici Hopf algebroid, namely the case where the coinvariants of the cleft extension coincide with the base of the Hopf algebroid, is a Drinfeld twist of a Hopf algebroid by a two-cocycle, generalizing work of Böhm, Han and Majid.

The matrix Hilbert series of a locally finite elementary twisted Calabi–Yau algebra is the inverse of a matrix polynomial. The Smith normal form of this polynomial over the power series ring at $x=1$ produces a finite list of local exponents refining the Gelfand–Kirillov dimension, which records only the largest of them. We show that the Calabi–Yau symmetry makes this local data rigid: the algebra decomposes into ring factors according to the average Artin–Schelter index along Nakayama cycles, and after a local normalization the symmetry induces a nonsingular linking form on the Smith cokernel together with a finite-order Nakayama action on its layers, forcing reciprocal-eigenvalue and parity constraints on the multiplicities of the exponents. We compute the complete local data for cyclic skew-group algebras, derive a parity sieve for quiver classifications in dimension three, and realize, as an iterated smash product of a graded down-up algebra, a four-vertex type that a recent classification had left open.

In this paper, we investigate certain properties on the Mittag-Leffler conditions via set-theoretic methods. We establish that a strongly $\aleph_1$-presented module $M$ satisfying Ext$_R^{\ge 1}(M,D^{(\aleph_1)})=0$ belongs to the left orthogonal class of $\overline{D}$, where $\overline{D}$ is the definable of $D$. This yields the consequence that every $\aleph_1$-generated strongly Gorenstein projective module is Gorenstein flat. Furthermore, we investigate when a flat Gorenstein projective module is projective. Finally, for any two-sided $\aleph_1$-coherent ring $R$, we prove the identity $\operatorname{silp}R+\operatorname{silp}R^{\mathrm{op}}=\operatorname{spli}R+\operatorname{spli}R^{\mathrm{op}}$.

In this work we investigate when an $n\times n$ Jordan matrix can be written as the sum of two strictly $n$-zero matrices. In particular, we show that if $\mathbb{F}$ is an algebraically closed field of characteristic zero and $A$ is an $n\times n$ matrix over $\mathbb{F}$ with $\tr(A)=0$, then $A$ is the sum of two strictly $n$-zero matrices.

We construct a simple $\lambda$-double Lie algebra on $k[t,t^{-1}]$ for every nonzero $\lambda\in k$. We then show that the analogous two-sided construction of weight zero is also simple. Both products admit natural coefficient realizations via row-and-column-finite operators associated with the finitary $\mathbb Z\times\mathbb Z$ matrix algebra.

We give an example of a finite ring for which the set of polynomials inducing the zero function fails to be a two-sided ideal, disproving a conjecture of Werner.

We study integer-valued polynomials and null polynomials over structural matrix rings, that is, rings whose elements are matrices in which an entry may be nonzero only when its row index precedes its column index in a fixed preorder. We prove that the integer-valued polynomials over a structural matrix ring with entries in an integral domain form a ring, and that the null polynomials over a structural matrix ring with entries in an arbitrary commutative ring form a two-sided ideal. In both settings, we give a characterization of the corresponding polynomials with matrix coefficients in terms of scalar-coefficient polynomials. These results extend corresponding theorems for full matrix rings and upper triangular matrix rings.

We study the relation between semisimple algebras, artinian simple algebras, and artinian absolutely flat algebras in ind-completions of tensor categories (rigid abelian monoidal categories). Our main results show that the three types of algebras coincide, if we restrict to commutative algebras in certain (all, if we accept the main conjectures in the field) symmetric tensor categories of moderate growth. We also provide examples to show that in general these notions tend to diverge, contrary to the classical case of algebras over a field. In one appendix we give the first, to the best of our knowledge, example of a tensor category with finite-dimensional morphism spaces but objects of infinite length. In a second appendix we establish a generalisation of Deligne's theorem that tannakian categories are neutral over algebraically closed fields.

We describe all injective Rota-Baxter operators $R$ of weight 1 on the polynomial algebra $F[x]$. When $\mathrm{char}\,F = p>0$, the only one is $R=-\mathrm{id}$. When $\mathrm{char}\,F = 0$, we have either $R = -\mathrm{id}$ or, up to conjugation with automorphisms of $F[x]$, $R(1) = x$, and $R$ is uniquely defined via this equality. Together with the known weight-zero case, this completes the classification of injective Rota-Baxter operators of any weight on $F[x]$.

For every Lie–Rinehart algebra, we construct an $F$-manifold algebra on the direct sum of its base algebra and module. Contrary to the assertion in \cite[Proposition 13.3.26]{LodayVallette}, the resulting structure is generally not Poisson. We determine when powers of the positive-degree ideal in the associated symmetric Poisson algebra are Poisson ideals, and relate the Leibnizator to the Lie–Rinehart differential. For a finite projective module of constant rank, the trace of the Leibnizator recovers the anchor and yields a rigidity result for injective anchors. We conclude with algebraic and geometric examples.