Papers

Let $R=L_{\F_2}(1,2)$. We prove that $\GL_r(R)$ is integrally acyclic for every $r\geq1$ and that the canonical map $\St_r(R)\to\GL_r(R)$ is an isomorphism for every $r\geq3$. The unit group $R^\times$ is finitely presented, and we describe an explicit finite presentation. The homology calculation uses leaf coordinates, simultaneous extensions of ordered frames, and finite-field actions on stabilizers. The Steinberg argument lifts relations from a simply connected frame complex and refines coordinates. We also state separate criteria for the two arguments over rings of characteristic two.

Given a matched pair of Hopf algebras, the double cross product construction yields a new Hopf algebra, the prototypical example of which is the generalized quantum double arising from a (Hopf) skew pairing. In this paper, we specifically investigate double cross products with projections in the framework of Radford's biproduct theory, and establish that such an algebraic structure is equivalent to a simplified version of (Yetter-Drinfeld) relative Rota-Baxter operators on Hopf algebras recently introduced by Sciandra, with the corresponding braided Hopf algebras lying in the Yetter-Drinfeld module category under coadjoint coactions. We further examine conditions under which this algebraic structure induces a matched pair of actions on a single Hopf algebra, which is in turn equivalent to the notion of braiding operators on Hopf algebras that yield solutions of the braid equation. In addition, we show that such relative Rota-Baxter operators are morphisms between certain Doi-Hopf modules.

Let $A=\Phi\langle X\rangle$ be a free associative algebra over a field of characteristic zero, and let $J$ be the Jordan subalgebra of $A^{(+)}$ generated by $X$ and $1$. For every $n\geq1$ we construct an element $U_n\in\mathbb Q[\mathfrak S_n]$ whose image on $A_n$ is exactly $J_n$. If \[ \det(tI-U_n|_{V_n})=t^{e_n}q_n(t),\qquad q_n(0)\ne0, \] on the multilinear component $V_n$, then \[ a\in J_n\quad\Longleftrightarrow\quad a\,q_n(U_n)=0. \] Thus the criterion gives a finite algorithm for recognizing Jordan elements: in each degree one constructs $U_n$ and $q_n$ and tests the single equation $a\,q_n(U_n)=0$. Moreover, $I-q_n(U_n)/q_n(0)$ is a projection of $A_n$ onto $J_n$. We give the projection explicitly in degrees at most four, record the multilinear dimensions through degree eight, and formulate the analogous criterion on each fixed multihomogeneous component.

We develop a general characteristic-free framework for studying the homological properties of a broad class of module-finite Hopf algebras. This framework makes it possible to reduce the study of the homological properties of many quantum groups at roots of unity and their multiparameter deformations to the study of the corresponding small quantum groups. For any affine Hopf algebra $H$ admitting a large central Hopf subalgebra $C$, we prove that its left homological integral space, in the sense of Lu-Wu-Zhang, is isomorphic as a bimodule to the left integral space of the identity fiber algebra, which is a finite-dimensional Hopf algebra. Consequently, $H$ is a symmetric Frobenius extension of $C$ if and only if the corresponding identity fiber algebra is unimodular and the square of the antipode of $H$ is inner, thus providing an effective criterion for the Calabi-Yau property of $H$. For a broad class of quantum groups at roots of unity, an appropriate large central Hopf subalgebra can be chosen such that the identity fiber algebra is the corresponding small quantum group. Therefore, some homological properties of these big quantum groups are governed by those of their corresponding small quantum groups. Assuming that the base field is algebraically closed, we prove that $H$ is unimodular if and only if, for some (equivalently, every) maximal ideal $\mathfrak{m}$ of $C$, the category of finite-dimensional representations of the fiber algebra at $\mathfrak{m}$ is unimodular in the sense of Yadav as a module category over the finite tensor category of finite-dimensional representations of the identity fiber algebra. As an application, we prove that all Andruskiewitsch-Angiono-Yakimov large quantum groups are affine noetherian unimodular Artin-Schelter Gorenstein Hopf algebras. We also give a necessary and sufficient condition for these large quantum groups to be Calabi-Yau.

We present an explicit infinite equational basis for the six-element additively idempotent semiring $TR_6$ and prove that $TR_6$ is nonfinitely based. We also give a complete description of the subvariety lattice of the variety generated by $TR_6$, showing that it forms a four-element chain. Our results demonstrate that the variety generated by $TR_6$ is a limit variety: it is itself nonfinitely based, yet all of its proper subvarieties are finitely based. This provides a new limit variety of additively idempotent semirings, distinct from all previously known ones. In fact, $\mathsf{V}(TR_6)$ is the first explicit limit subvariety of the variety generated by the max-plus algebra $\mathbf{N}$. Moreover, $TR_6$ is not strongly nonfinitely based: it belongs to a finitely based variety generated by a finite additively idempotent semiring. Together with the six-element additively idempotent semiring $SR_6$, these are the first two finite additively idempotent semirings that are nonfinitely based but not strongly nonfinitely based. Finally, we study the variety generated by $SR_6$ and $TR_6$, showing that it is nonfinitely based and has exactly nine subvarieties, four of which are nonfinitely based and the remaining five are finitely based.

The thesis develops symbolic representations of dynamical systems in finite-dimensional commutative real algebras. The algebraic framework (orthogonal idempotents, zero divisors, and the structure of finite-order algebras) supports algebra-valued Fourier and Laplace transforms and Walsh-function-based system identification. The methods yield exact, coordinate-free vectorial solutions for motion in non-inertial reference frames, including a generalized Larmor theorem, the Foucault pendulum, motion in central positional-force fields, and the two-body problem in arbitrarily rotating reference frames. This English edition is an audited LaTeX reconstruction of the 1995 original, with an editorial note documenting provenance.

We construct a counterexample to Köthe's conjecture and use this to also answer a question due to Rowen.

Gheibi, Jorgensen and Takahashi recently introduced the quasi-projective dimension, a homological invariant that extends the classical projective dimension. In this paper, we investigate this invariant from the perspective of Gorenstein homological algebra. First, we show that the quasi-projective dimension coincides with the Gorenstein projective dimension under certain conditions. Second, we introduce and study the quasi-Gorenstein projective dimension as a Gorenstein analogue of the quasi-projective dimension.

In these lecture notes, we review the recent results around the chain rule for the generalised KPZ equation and the geometric KPZ equation. We introduce the main ideas of Regularity Structures via a B-series type expansion and the two main combinatorial sets for encoding this expansion: decorated trees and multi-indices. Then, we explain how the chain rule was first understood via integration by parts and diagrammatic computations. For the generalised KPZ equation, a scalar-valued equation, we use multi-indices to derive a characterisation in terms of iterations of covariant derivatives. The same characterisation for decorated trees is more involved and requires the use of operadic and homological tools. We summarise the main arguments of this proof.

Let $n\geq 3$, let $H\subseteq\mathbb{C}^n$ be any additive subgroup spanning $\mathbb{C}^n$, and let $0\ne t\in H$. We study Pozhidaev's central simple $n$-Lie algebra $P(H,t)=\widetilde{\mathcal A}(H,t)/\mathbb{C}e_0$ without a finite generation or discreteness assumption on $H$. Its inner derivation algebra is the simple generalized divergence-free Lie algebra $\mathcal S(0,0,n;t,H)$. We prove that its space of inner-equivariant symmetric products vanishes and that every $1/n$-derivation is a scalar multiple of the identity. Using these invariants, we show that $P(H,t)$ is not isomorphic to any simple nonabelian $n$-Lie algebra defined on the underlying spaces of the $S$, $W$, or $SW$ constructions recorded by Cantarini and Kac. We also show that $P(H,t)$ is the quotient by the constants of the derived algebra of an explicit $S$-algebra on $\mathbb{C}[H]$. Finally, we realize Pozhidaev's second construction $E(H)$ over $\mathbb{C}$ as a $W$-algebra.

We review some topics in "exceptional mathematics” from the perspective of 3-dimensional geometry: the octonions $\mathbb{O}$, the split octonions $\mathbb{O}'$, the bioctonions $\mathbb{O}_\mathbb{C} \cong \mathbb{C} \otimes_\mathbb{R} \mathbb{O}$, the complex Albert algebra $\mathfrak{h}_3(\mathbb{O}_\mathbb{C})$, and the complex form of the exceptional Lie algebra $\mathfrak{e}_6$. We show how to functorially build an algebra isomorphic to $\mathbb{O}$ from any 3d complex vector space equipped with an inner product and complex volume form. Similarly, we build one isomorphic to $\mathbb{O}'$ starting from a 3d real vector space equipped with a volume form, and one isomorphic to $\mathbb{O}_\mathbb{C}$ starting from a 3d complex vector space equipped with a complex volume form. We give applications to 3-dimensional real and complex manifolds. Finally, we describe how to build an Jordan algebra isomorphic to $\mathfrak{h}_3(\mathbb{O}_\mathbb{C})$ starting from three 3d complex vector spaces equipped with complex volume forms. This last construction gives a nice explicit description of the complex Lie algebra $\mathfrak{e}_6$ and its subalgebra $\mathfrak{sl}(3,\mathbb{C}) \oplus \mathfrak{sl}(3,\mathbb{C}) \oplus \mathfrak{sl}(3,\mathbb{C})$.

The purpose of this paper is to classify all the Rota-Baxter operators explicitly on Grassmann algebra in dimension four over the reals.

It is well known that the polymorphism clone of a quasiorder is determined by its unary part, in the sense that any operation is a polymorphism if it satisfies the condition that all unary functions obtained by fixing some of its variables at constant values are polymorphisms. We show that a clone satisfies an analogous property for a particular fixed arity $d$ if and only if it is determined by a collection of what we call higher-dimensional generalized quasiorders whose dimension is equal to $d$. We define higher-dimensional generalized quasiorders with the appropriate generalizations of reflexivity and transitivity to the higher-dimensional setting, which rely on an explicit analysis of 'rectangular' arity.

We give a \(3\times 3\) counterexample, valid over every field, to a rank-minimization conjecture of Lin and Wimmer (Bull. Aust. Math. Soc., 84 (3) (2011), 441–443) related to Roth's similarity theorem for the Sylvester matrix equation. We also prove that, over the complex field, no counterexample can occur when one of the two matrix sizes is less than \(3\). Hence the example is dimensionally minimal over the complex field.

We introduce the notion of perfect regularity as an appropriate weakening of perfect normality, both for spaces and locales. Various characterizations are given, using Dedekind-MacNeille completions, injective hulls, and sublocales. We place the new class of perfectly regular frames among various well-studied classes of frames. We also introduce the construction of perfect regularization of a completely regular frame, compare it to Isbell's well-known booleanization construction, and argue that it is at least as important as the latter.

We introduce $\delta$-mock-Novikov algebras as a parameter-dependent mock analogue of Novikov algebras. For $\delta=1$, the commutator of a mock-Novikov algebra defines a Malcev algebra. Every $\delta$-mock-Novikov algebra satisfying $\mathcal{A}^2\subseteq\operatorname{Ann}(\mathcal{A})$ is shown to be differentially special. At the operadic level, the binary quadratic operad governing $\delta$-mock-Novikov algebras is quadratically self-dual but not Koszul. We prove that finite-dimensional $\delta$-mock-Novikov algebras are nilpotent in characteristic zero and classify those of dimension at most four. Finally, we study the associated Poisson-type structures and establish relations and constructions among them via polarization, depolarization, and tensor products.

We classify all filtered deformations of polynomial rings in three variables. We use our characterization to answer a special case of a question of Etingof by showing that if $F$ is an algebraically closed field of positive characteristic and $A$ is an $F$-algebra with a filtration by finite dimensional subspaces whose associated graded ring is a polynomial ring in three variables then $A$ satisfies a polynomial identity.

Iima and Yoshino asked for an ideal $I$ in $S=k[x_1,x_2,\ldots]$, with $\operatorname{deg} x_i=i$, and a monomial order such that $S/I\cong k[x_i:i\equiv\pm1\pmod5], \operatorname{in}(I)=(x_i^2,x_ix_{i+1}:i\geq1).$ We construct such an ideal and monomial order over every field $k$ of characteristic different from $5$ containing an element $c$ with $c^2+c=1$. The ideal has an explicit infinite homogeneous reduced Gröbner basis. A five-periodic syzygy derived from a pentagon identity proves that all basis relations belong to $I$ and supplies standard representations for the non-coprime critical pairs. Triangular elimination establishes the graded quotient isomorphism. Together, the quotient and initial ideal descriptions yield the partition form of the first Rogers-Ramanujan identity. In each weighted degree, a perfect matching in the support of the normal-form matrix gives a bijection between the two partition classes. We formalize the complex specialization in Lean 4 using Mathlib and our set-based theory of infinite Gröbner bases, including the reduced basis, the graded quotient isomorphism, and the partition-matching theorem.

We study slice regular functions on the real division algebras $\mathbb{H}$ and $\mathbb{O}$ from the point of view of equivalence under automorphism and conjugation actions. Motivated by recent orbit-theoretic descriptions in terms of the invariants $(Tr,N,cdiv)$, and by the quaternionic theory of $*$-conjugation by semiregular functions, we investigate whether a slice regular function can be replaced by a canonical representative in its equivalence class. The normalization considered in this paper is algebraic: we look for representatives whose isolated nonreal zeros are aligned in a single complex slice. We call such a representative a model when it belongs to the strong-equivalence class of the original function. We prove that purely vectorial functions always admit models, while in general strong equivalence is too rigid and such representatives need not exist. We therefore introduce semi-models, which preserve the symmetrization and the real and spherical part of the zero set, but are allowed to leave the original strong-equivalence class. Our main result proves that every slice regular function $f$ with $N(f)\not\equiv 0$ admits a semi-model. The construction clarifies the different roles of the invariants $(Tr,N,cdiv)$ and the distinction between strong equivalence, weak equivalence, and semiregular conjugacy.

We prove a Snake Lemma for 2-categories. Working in a 2-category with a strong bizero object, we develop 2-kernels and 2-cokernels, 2-monomorphisms and 2-epimorphisms as fully faithful and cofully faithful 1-cells, short 2-exact sequences and normal image factorisations, and we prove a two-dimensional Normal Short Five Lemma. We then introduce the dinversion of an antinormal pair and the notion of a homologically self-dual 2-category, characterised equally by the self-duality of homology, by a Pure Snake Lemma and by a Third Isomorphism Property. Two-dimensional di-exactness implies homological self-duality. Our main result is the Snake Lemma in a 2-di-exact 2-category: a ladder of 2-exact rows with normal verticals induces a 2-exact six-term sequence, with a connecting 1-cell that is 2-natural in the ladder. Di-exactness can be traded for two hypotheses that are not self-dual: that dinversion preserve normality, and that normal 2-epimorphisms compose. Everything specialises, on passing to a locally discrete 2-category, to its classical counterpart. We close by exhibiting three models: a 2-di-exact 2-category of abelian categories containing $\mathsf{Coh}(X)$ for every noetherian scheme $X$; the locally ordered 2-category of complete modular lattices, in which 2-di-exactness amounts to Dedekind's transposition principle; and the 2-category of Hilbert lattices, which is not 2-di-exact but satisfies the non-self-dual hypotheses, by a theorem of Mackey on pairs of closed subspaces.

We study the generic nullity of generalized commutator operators \[L_{\mathbf{A}}(X)=s_{k+1}(A_1,\cdots,A_k,X)\] on matrix algebras, where $s_{k+1}$ denotes the standard polynomial. Dixon and Pressman conjectured an explicit formula for the generic nullity of $L_{\mathbf{A}}$, and Brassil and Reichstein proved the conjecture when $k$ is even. In this paper, we settle the remaining case where $k$ is odd. Our proof first treats the boundary cases $k=2n-3$ in dimensions $n$ and $n+1$ using degree decompositions and graph-theoretic interpretations of alternating trace forms, and then establishes a dimension-extension argument from $n$ to $n+2$. Consequently, together with the result of Brassil and Reichstein, we obtain a complete proof of the Dixon-Pressman generic nullity conjecture over any field of characteristic zero.

A convex semilattices is a convex algebra endowed with an additional semilattice operation that satisfies a distributive law. We investigate free convex semilattices by studying the homomorphisms into products R{\Omega} and, more specifically, the homomorphisms between two free finitely generated convex semilattices. We characterise the finitely generated subalgebras of a free finitely generated convex semilattice.

In this paper, first we give a new characterization of the cohomology of pre-Lie algebras using the Chevalley-Eilenberg cohomology associated to a morphism from the operad of Lie algebras to Hadamard product of the operad of pre-Lie algebras and its Koszul dual operad. Then we apply the same approach to study the cohomology of Novikov algebras, and give the cochain complex explicitly. The cochain complex of the underlying pre-Lie algebra is shown to be isomorphic to the quotient of the cochain complex of a Novikov algebra. Consequently, there is a long exact sequence connecting the cohomologies of a Novikov algebra and the underlying pre-Lie algebra. The cohomology of a Novikov algebra with coefficients in a representation is introduced using pseudo-tensor categories. As applications, we show that infinitesimal deformations and abelian extensions are classified by the second cohomology groups with different coefficients. Various examples are given to illustrate the difference between the cohomology of a Novikov algebra and that of the underlying pre-Lie algebra.

Matlis duality, first introduced by Eben Matlis in 1958, stands as one of the most elegant and powerful tools in commutative algebra, establishing a striking contravariant equivalence between Artinian and Noetherian modules over complete local rings. In this paper, we undertake a thorough and systematic investigation of Matlis reflexive modules — those modules for which the canonical evaluation homomorphism into the double Matlis dual is an isomorphism. Our exposition begins with a detailed historical narrative, tracing the intellectual trajectory from classical Pontryagin duality through Grothendieck's dualizing complexes to the modern formulation of Matlis duality. We then develop the core theory from first principles, demonstrating that the class of Matlis reflexive modules constitutes a Krull-Schmidt category — a result that generalizes the classical decomposition theorems for modules of finite length. Over complete Noetherian local rings, we prove that Matlis reflexive modules coincide precisely with the class of minimax modules (those possessing a Noetherian submodule whose quotient is Artinian). We establish full closure properties, including stability under submodules, quotients, extensions, and finite direct sums, with rigorous and self-contained proofs. The paper also surveys recent applications to generalized local cohomology, change-of-rings results, and connections to pure-injective modules and linear compactness, and concludes with a discussion of open problems and future research directions.

We introduce a notion of $2$-homological category modelled on pointed homological categories in the sense of Grandis, using bizero objects whose null $1$-cells are zero objects in the hom-categories for formulating $2$-dimensional pointedness. We first prove directly that the $2$-category of di-exact homological categories, which generalize Puppe exact categories, functors preserving all kernels and cokernels, and arbitrary natural transformations is $2$-homological. Its normal subcategories are saturated thick subcategories, and its exact quotients are constructed by a complete three-arrow fraction calculus similar to the one known for Puppe exact categories. We then adapt this proof to prove that the $2$-category of triangulated categories is also $2$-homological; here, instead of the ternary fractions we use the well-known Verdier fractions. Abstracting the common quotient structure of these proofs yields a general criterion, using which we further establish that the $2$-categories of pointed, additive and abelian categories are also $2$-homological. The criterion also applies to categories enriched in semimodules over a fixed commutative rig, with a zero object, and to their full sub-$2$-category with finite biproducts. Their quotients are linear congruence quotients. Combining enrichment with Puppe exactness gives further examples whose quotients are exact linear localizations, including linear abelian categories as the finite-biproduct case. In the abelian and triangulated cases the normal subcategories and quotients are, respectively, Serre subcategories and Serre quotients, and thick triangulated subcategories and Verdier localizations.

We characterize the arithmetical categories - the exact Mal'tsev categories whose lattices of congruences are distributive - by a simple condition on some suitable pushouts and a finite limit: a regular category is exact arithmetical if and only if, for every triple of regular epimorphisms with common domain, the pairwise pushouts exist and the comparison morphism to the limit of the diagram they form is a regular epimorphism. For a regular category, the validity of this condition for pairs and for triples is equivalent to its validity for $n$-tuples for all $n \ge 2$, and characterizes arithmetical categories among regular ones. For pairs alone the condition reduces to the characterization of exact Mal'tsev categories among regular ones, due to Carboni, Kelly and Pedicchio. For $n$-tuples with $n \ge 4$ nothing further is obtained: arithmeticity is the third and last rung of that ladder. The proof rests on a categorical form of the Chinese Remainder Theorem; in the exact Mal'tsev context this form of the theorem lies close to results of Hoefnagel on majority categories. As a direct application, we give a new proof of Pixley's characterization of arithmetical varieties using a suitable diagram of free algebras.

Slice regular functions are a generalization of holomorphic functions where alternative real $*$-algebras are considered instead of the field of complex numbers. For such functions we show that zero sets and function values on suitable subsets may be prescribed. As a consequence, we show that for any axially symmetric domain there exist slice regular functions which (due to the nature of its zero set) can not be extended to a larger such domain.

We introduce notions of integrality and full integrality over one-sided ideals, together with associated radical constructions, and develop a general framework for their study. As applications, we extend several classical results concerning Jacobson radicals to the one-sided setting. In particular, we prove that the Jacobson radical of a left ideal is integral over that ideal in left artinian rings and in algebraic algebras, obtaining a one-sided analogue of a theorem of Amitsur. We also show that Jacobson radicals of graded left ideals are graded and derive a corresponding extension of Amitsur's theorem on Jacobson radicals of polynomial rings.

We study finite-sample rigidity for left translates of similarity orbits over commutative rings. Let \(R\) be a commutative ring with \(2\in R^\times\), and let \(C\in\operatorname{Mat}_2(R)\) be cyclic. Given matrices \(G_i=FK_i\), where \(F\in\operatorname{GL}_2(R)\) is common and the underlying points \(K_i\) lie in \(\operatorname{Orb}(C)\), we determine when a finite translated sample has exactly the same multiplier ambiguity as the complete translated orbit. The key structural result is that the orbit differences span \(\mathfrak{sl}_2(R)\), which yields a scalar description of the left stabilizer. We then give a determinantal saturation criterion, expressed directly in the matrices \(G_i\), under which finite-sample consistency reduces to trace equations and one determinant condition. Off-diagonal quadratic operators on pairs of free rank-two modules realize the translated-orbit model exactly. Over fields we obtain sharp sample thresholds under the saturation criterion, and over finite fields we derive exact saturation probabilities for regular semisimple and nonzero nilpotent orbits. Torsion restrictions of elliptic isogenies provide a natural arithmetic realization of the rank-two module framework.

For an $n\times n$ oscillatory matrix $A$, let $e_k(A)$ be the least positive integer $m$ for which every minor of order $k$ of $A^m$ is positive. We determine the profile $(e_1(A),\ldots,e_n(A))$. For every nonsingular totally nonnegative matrix, the column index sets of positive entries in each compound row form an interval in the componentwise order. The endpoint maps are order-preserving and compose under multiplication. This yields a two-corner criterion for positivity of all minors of a fixed order. Consequently, $e_k(A)$ is the larger of the first positivity times of two remote corner minors. We prove the sharp inequalities $e_k(A)\leq\max{k,n-k}$ for $1\leq k<n$ and $|e_{k+1}(A)-e_k(A)|\leq1$ for $1\leq k\leq n-2$. For $D=\operatorname{diag}(1,-1,1,-1,\ldots)$, the matrix $DA^{-1}D$ is oscillatory and satisfies $e_k(DA^{-1}D)=e_{n-k}(A)$ for $1\leq k<n$; the determinant exponent remains one. The ordered positions of the positive factors in an adjacent bidiagonal factorization determine the profile independently of their values. Each one-sided factor sequence can be represented by a single permutation, giving a finite characterization of all profiles and an integer unimodular realization of each. We also obtain a compatibility condition across minor orders: if $3\leq k\leq n-3$ and $e_k(A)\leq2$, then $e_j(A)\leq2+\lceil |j-k|/2\rceil$ for $2\leq j\leq n-2$. In particular, $e_k(A)\leq2$ implies $e_{k+2}(A)\leq3$ for $3\leq k\leq n-4$, and both the constant and this range are sharp.