Papers

We show that the six-element additively idempotent semiring $B_2^1$ is inherently nonfinitely based. More precisely, for every positive integer $n$, we construct an infinite additively idempotent semiring, generated by at most $(6n+8)^2$ elements, that satisfies every $n$-variable identity of $B_2^1$.

We study the Ooms multiplicities of Frobenius maximal parabolics $W(a,b)$ of type~$A$. For $a=mb+r$ with $1\le r<b$, the Euclidean algorithm relates their potential histograms to those of the smaller algebra $W(r,b-r)$. We prove strict unimodality for every positive coprime pair: the multiplicities increase strictly to the equal central values at eigenvalues $0$ and $1$, then decrease strictly. This strengthens the recent maximal-parabolic unimodality theorem of Giaquinto, Irving, Lauve, and Mastnak. We also prove log-concavity for the unbounded families $W(mb\pm t,b)$, where $t\in\{1,2,3\}$, $b>t$, $\gcd(b,t)=1$, and $m\ge1$. These results cover every Frobenius case with smaller block at most $8$, and also smaller block $10$. For each fixed residue class, we then prove that log-concavity of the whole family is determined by finitely many initial spectra. The finite cutoff is obtained symbolically. Exact integer verification of the resulting finite lists proves strict internal log-concavity whenever the smaller block is at most $128$, with no bound on the larger block. The unrestricted log-concavity conjecture is not proved here.

We study integral dependence over left ideals using G-left ideals, namely left ideals maximal with respect to the failure of integrality of a fixed element. We establish several characterizations of G-left ideals, and show that their intersection is precisely the upper nilradical. Motivated by the Köthe conjecture, we introduce the class of left KI-rings as rings in which sums of left ideals integral over a fixed left ideal remain integral over that ideal. It is shown that left artinian rings, PI-rings, algebraic algebras, and algebras of dimension smaller than the cardinality of the ground field are left KI-rings.

Using Galois covering theory and equivariant techniques, we classify all connected representation-tame Geiß–Leclerc–Schröer (GLS) algebras in terms of their defining triples $(C,D,\Omega)$. We then study the GLS algebras $H(\widetilde{CD}_n)$ with minimal symmetrizers, where $n\geq 2$ and $\widetilde{CD}_2=\widetilde{B}_2$. By realizing these algebras as basic algebras of $\mathbb{Z}_2$-skew group algebras of the string algebras $H(\widetilde{C}_{2n-2})$, we introduce extended strings and extended bands to parameterize the connected components of the Auslander–Reiten quivers of $H(\widetilde{CD}_n)$ and determine their shapes. We further classify the indecomposable $\tau$-locally free modules over representation-tame GLS algebras of affine type and show that their rank vectors form precisely the set of positive roots together with explicitly described non-root vectors in the positive cone of the root lattice. This yields a precise revision of the GLS conjecture for representation-tame GLS algebras of affine type.

Let $\Gamma$ be a coalgebra over a field. There is a well-known covariant embedding of the category of graded $\Gamma$-comodules into the category of graded modules over the dual algebra $\Gamma^*$. We show that this covariant embedding preserves Ext-groups in a range of motivating cases, including in some cases like Steenrod algebras where the covariant embedding has otherwise bad homological properties and fails to preserve Ext-groups on the underlying ungraded categories. As an application, we prove the old conjecture in stable homotopy theory that there are no nonzero projectives in the category of graded comodules over the dual Steenrod algebra. As another application, we carry out a study of the right adjoint to the extended comodule functor, including a topological interpretation of this right adjoint. This yields generalizations of formulas of Margolis and Lin characterizing maps from Eilenberg-Mac Lane spectra into finite-type bounded-below spectra; our generalizations drop the "finite-type" hypothesis.

A Kähler structure on a real Lie algebra $\mathfrak{g}$ is a pair $(\omega, J)$ of a 2-cocycle and a complex structure on $\mathfrak{g}$ which are compatible in the sense that $J$ is skew-symmetric with respect to $\omega$. This paper investigates Kähler structures on Lie algebras, with a particular focus on the nilpotent case. We introduce an algebraic framework for the Kählerian reduction and oxidation of Lie algebras by one-dimensional and two-dimensional ideals. Our main structural result demonstrates that every indecomposable, quasi-nilpotent Kählerian nilpotent Lie algebra of dimension greater than four can be inductively reconstructed through a finite sequence of Kählerian central oxidations by planes. This reduction sequence originates from a base algebra that is either $\{0\}$, $\mathbb{R}^2$, or one admitting a strongly non-nilpotent complex structure. Moreover, if the complex structure is nilpotent, all intermediate reductions also have a nilpotent complex structure.

It is known that the inverse of an invertible real square matrix satisfying the treeangle property, is a treediagonal matrix. A converse statement also holds. We show that the verbatim analogues are not true for the Moore-Penrose inverse, and obtain the precise structure of graphs corresponding to the Moore-Penrose inverse of matrices possessing the treeangle property.

We study $n$-torsionfree objects in abelian categories with enough projectives. Frobenius functors preserve $n$-torsionfreeness, and faithful ones reflect it. We prove that stabilization of the torsionfree filtration implies weak Gorensteinness. For Frobenius extensions satisfying a generator condition, we compare the terms of minimal injective resolutions and obtain transfer of Auslander-type conditions and of the Auslander–Gorenstein conjecture. We also compute a family of non-Gorenstein algebras whose torsionfree filtrations stabilize at level two and contain explicit nonprojective Gorenstein projective modules.

Let $\mathbb{R}\langle X,Y\rangle$ be the free algebra on two symmetric letters. A polynomial is trace-positive if it has non-negative trace on every pair of real symmetric matrices, and it lies in the cone $\Theta$ if it is cyclically equivalent to a sum of Hermitian squares. Burgdorf and Klep proved that the two notions coincide in degree at most four; from degree six onwards they differ. We give a complete, explicit description of both cones on the six-dimensional space $Z$ of symmetric homogeneous sextics in two symmetric letters whose abelianization vanishes identically, and show that on $Z$ they coincide: for $f\in Z$, trace-positivity is equivalent to membership in $\Theta_3$, to non-negativity of the trace on pairs of $2\times2$ symmetric matrices, and to being cyclically equivalent to a sum of four Hermitian squares of cubics with vanishing abelianization. Membership in $\Theta$ reduces to a $4\times4$ positive semidefinite Gram matrix whose diagonal blocks are fixed by the content-$(4,2)$ and $(2,4)$ parts of $f$; each of those parts ranges over a two-dimensional cone with two linear facets, and the content-$(3,3)$ part is then confined to a rectangle whose sides are the geometric means $\sqrt{p_+q_+}$ and $\sqrt{p_-q_-}$ of the eigenvalues of those blocks. Each facet is certified by one of two families of pairs of $2\times2$ symmetric matrices. The abelianization hypothesis is natural: every bivariate degree-six example we are aware of – a trace-positive polynomial outside $\Theta$ – arises by lifting a non-negative, non-SOS polynomial in two variables to a cyclically sorted polynomial; and that lifting cannot operate when the abelianization vanishes. Symmetry of $f$ is essential: dropping it adds one antisymmetric direction, on which the trace vanishes identically, yet whose non-zero elements all lie outside $\Theta$.

Burde proved that the Lie algebra $\mathfrak{sl}_2(\mathbb{K})$, over a field $\mathbb{K}$ of characteristic $p > 2$, admits compatible left-symmetric algebra structures if and only if $p = 3$. Since the even part of the Lie superalgebra $\mathfrak{osp}(1|2)$ is isomorphic to $\mathfrak{sl}_2(\mathbb{K})$, we ask whether Burde's structures can be extended to compatible left-symmetric superalgebra structures on $\mathfrak{osp}(1|2)$. We verify Burde's left-symmetric structures on $\mathfrak{sl}_2(\mathbb{K})$, and then test their possible extensions to $\mathfrak{osp}(1|2)$. We show that no such extension exists in characteristic 3. Together with the obstruction coming from the even part for characteristic $0$ and $p > 3$, this gives non-existence for a field $\mathbb{K}$ of characteristic different from $2$.

We introduce a bi-oriented interlaced matrix product on $2\times 2$ matrices over a unital, possibly non-associative algebra, extending a matrix mnemonic for Cayley–Dickson doubling. The opposite orientations $X_R$ and $X_L$ coincide precisely when the underlying algebra is commutative. For the canonical position and momentum operators $Q,P$ on $L^2(\mathbb{R})$, a matrix $D$ and its companion $\widetilde{D}$ satisfy, on the common Schwartz domain, $\widetilde{D} X_R D =\widetilde{D} X_L D =2H Id_2$, where $H=(P^2+Q^2)/2$. We study the associated interlaced eigenvalue equations for operator-valued columns. The combinations $\Phi=A+iB$ and $\Psi=A-iB$ decouple the equations into two-sided Sylvester equations. We construct explicit rank-one generalized eigenoperators from position and momentum eigenstates. Their distributional kernels belong to the doubled rigged Hilbert space $S(\mathbb{R}^2)\subset L^2(\mathbb{R}^2) \subset S'(\mathbb{R}^2)$; the corresponding position and momentum operators act on different tensor factors and therefore commute. The rank-one families do not exhaust the distributional solutions: higher-order generalized solutions are also exhibited. We distinguish the algebraic identities on the Schwartz core from questions concerning closed-operator realizations.

Assuming the nonexistence of uncountable measurable cardinals, we construct an $\mathbb F_2$-algebra and a covering class of right modules which is not closed under colimits of countable chains. This gives a counterexample to Enochs' conjecture under $V=L$.

Let $K$ be an infinite field of characteristic $2$. It is well known that the Lie algebra $\mathfrak{gl}_2(K)=M_2(K)^{(-)}$ of all $2\times 2$ matrices does not admit a finite basis of polynomial identities. Similarly, the finite basis problem for the associative algebra $M_2(K)$ remains open. In this note, we obtain two multilinear identities of degree $5$ for $M_2(K)$ and prove that, together with the standard polynomial of degree $4$, they generate all multilinear identities of degree up to $7$. Moreover, we prove that none of the three generators can be omitted.

The multiplication operator of an idempotent in a metrized commutative nonassociative algebra is a self-adjoint linear endomorphism whose spectrum reflects the algebraic and geometric structure encoded in the algebra. Although for general algebras the spectra of idempotents can behave arbitrarily, for the most interesting classes – among them algebras of Clifford and Jordan type arising in geometry – the spectrum is highly constrained. We consider a family of algebras determined by a partial Steiner triple system (PSTS) with blocks decorated by signs, each block determining an idempotent; even for the simplest PSTS the resulting algebras can be quite complicated. For decorations of the Grassmannian PSTS $G_2(m)$, whose blocks are $3$-subsets of an $m$-element set and whose points parametrize an underlying incidence geometry, the spectra behave rigidly: the spectrum with multiplicity of a block idempotent is a function of a single combinatorial parameter, the number of negatively signed Pasch configurations – quadrilateral subconfigurations – containing the block. The resulting block spectra organize into a spectral profile, recovered by binomial inversion from a hierarchy of moment invariants. For a distinguished family of decorations, the associated algebras are axial algebras satisfying a fusion law independent of $m$, and for these algebras there is given an alternative explicit matrix model exhibiting the rigidity directly; the even part of this model is a direct sum containing a polar algebra summand, linking the construction to polar algebras and symmetric Clifford systems.

We introduce semiconstellations and restriction semiconstellations, together with their actions on sets, providing a common setting for partial actions of categories, inverse semigroupoids, restriction semigroups, and semigroupoids. We then investigate the globalization problem, which asks whether every partial action can, up to isomorphism, be obtained by restricting a global action. A distinctive feature of our approach is the use of reflectors not only to construct globalizations, but also to study when partial actions are globalizable. We show that every partial action of a restriction semiconstellation admits a reflector in the full subcategory of $S$-algebras consisting of what we call almost global actions, and investigate conditions under which this reflector provides a globalization. As an application, we obtain a positive answer to the globalization problem for partial actions of restriction semigroupoids on sets, and consequently for restriction semigroups and restriction categories. In contrast, the answer is negative for restriction semiconstellations in general. Finally, given a semiconstellation $S$ and a suitable equivalence relation $R$ on $S$, we introduce the notion of an \emph{$R$-compatible action of} $S$ and construct a restriction semiconstellation $S^R$. We then show that partial and $R$-compatible actions of $S$ correspond, respectively, to partial and global actions of $S^R$. In this way, the globalization problem for semiconstellations can be reduced to the corresponding problem for restriction semiconstellations.

Radical truncation forgets information about a module. We study the ambiguity created by this loss: which nonisomorphic modules remain indistinguishable after all proper radical truncations have been fixed, how the resulting families are organized, and what information can be recovered from them. We introduce terminal ambiguity geometry to make this residual indeterminacy into a geometric object. The remaining ambiguity is governed by the terminal radical layer, and we classify terminal completions by Grassmannian orbit spaces, with projective ambiguity appearing as the rank-one case. This classification gives criteria for rigidity and nonuniqueness and, in the split case, determines the dimensions of the completion spaces and gives exact finite-field counts of nonisomorphic terminal completions. We also establish reconstruction and symmetry results for the rank-one projective geometry. Under suitable rank hypotheses, the ambiguity geometry recovers the corresponding residue division-algebra data, terminal species, and terminal radical layer, while its automorphisms are described by classical projective semilinear groups. Thus the paper studies both sides of radical truncation: the ambiguity produced by the information it forgets and the extent to which that information can be recovered from the resulting geometry.

In this paper, we study Zhang twist of Leavitt path algebra $L_K(E)$ using a graded automorphism $\sigma$ induced by a graph automorphism such that the twisted algebra is a Leavitt path algebra over another graph $E_{\sigma}$ which is a twisted graph obtained from the original graph $E$. We establish combinatorial connection between the graphs $E$ and $E_{\sigma}$ and as a consequence, we show that many ring-theoretic properties are invariant for Leavitt path algebras under the twist by $\sigma$. We also define the notion of noncommutative projective scheme for $\mathbb Z$-graded algebras that coincides with the notion of noncommutative projective scheme for connected $\mathbb N$-graded algebras defined by Artin and Zhang and study it in the context of twists of Leavitt path algebras.

Let $G$ be a group and $k$ be a commutative ring. The Gorenstein homological dimension $\text{Ghd}_k G$ and the Gorenstein cohomological dimension $\text{Gcd}_k G$ of $G$ over $k$ are defined as the Gorenstein flat and the Gorenstein projective dimension, respectively, of the trivial $kG$-module $k$. We prove that the Gorenstein cohomological dimension of a countable group $G$ is at most one more than its Gorenstein homological dimension and, more generally, that $\text{Gcd}_k G\le\text{Ghd}_k G+m+1$ for every $m\ge0$ and every group $G$ of cardinality at most $\aleph_m$. When the supremum sfli$k$ of the flat dimensions of the injective $k$-modules is finite, this gives $\text{Ghd}_k G\le\text{Gcd}_k G\le\text{Ghd}_k G+1$ for every countable group $G$, so that the two dimensions are finite simultaneously. These results are Gorenstein analogues of Bieri's inequality for the homological and the cohomological dimension of countable groups. They rest on a module-theoretic theorem of independent interest, which is a Gorenstein version of results of Jensen and Osofsky on the projective dimension of flat modules: over an arbitrary ring, every $\aleph_m$-presented Gorenstein flat module has projectively coresolved Gorenstein flat dimension at most $m+1$. An appendix characterizes the modules of PGF-dimension at most $n$ as the direct summands of the strongly $n$-PGF modules.

2-step ideals are ideals $I$ of the polynomial ring that satisfy $\mathfrak{m}^{k+2}\subsetneq I\subsetneq \mathfrak{m}^k$ where $\mathfrak{m}$ is the maximal ideal generated by variables. This class of ideals was recently introduced in [F. Giovenzana, L. Giovenzana, M. Graffeo, P. Lella: New components of Hilbert schemes of points and 2-step ideals, 2025, arxiv: 2507.02789] with the aim to obtain new irreducible components of $\mathrm{Hilb}^d(\mathbb{A}^n)$ and in particular to obtain new loci in $\mathrm{Hilb}^d(\mathbb{A}^3)$ of large dimension. In this paper we use the correspondence between Hilbert schemes and varieties of commuting matrices to define loci of commuting matrices that correspond to 2-step ideals. Then we estimate the dimensions of the obtained loci to get new proofs for the estimates of dimensions of loci of 2-step ideals in the case $n=3$. In this case we are also able to omit some assumptions in the dimension estimates in the above mentioned paper.

Let K be any field and E any directed graph. We investigate the existence of divisible non-injective left modules over the Leavitt path algebra L_K(E) for a broad class of graphs. Our main tool is a general criterion for divisibility that is particularly well suited to our setting.

In the path algebra $KE$, where $K$ is a field and $E$ is a directed graph (or quiver), every non-loop edge can be expressed in the form $arb$, where $a,b,r\in KE$ and $ab=0$. This fact leads us to the characterization that $KE$ is semicommutative if and only if $E$ contains no non-loop edges. For a ring $R$, let $Z_i(R)=\{x\in R: x=arb, a,b,r\in R, ab=0 \}$ and call elements of $Z_i(R)$ zero-insertive. It follows that $R$ is semicommutative if and only if $Z_i(R)\subseteq E(R)$ and weakly semicommutative if and only if $Z_i(R)\subseteq N(R)$. We establish that every non-unit element of the Leavitt path algebra $L_K(A_2)$ is zero-insertive. If $K$ is a field and $n\geq 2$ is an integer, then each zero-insertive element of $L_K(A_n)$ is nil-clean if and only if $K\cong \mathbb{F}_2$. We call a ring $R$ zero-insertive nil clean (ZINC) if every zero-insertive element is nil clean. We show that the Leavitt path algebra $L_{\mathbb{F}_2}(A_n)$ is a ZINC ring. Additionally, we investigate the behavior of zero-insertive elements and ZINC rings under various ring extensions.

Derivation algebras are natural structural invariants of Leibniz algebras. In a number of earlier papers, derivations were described for several classes of one-generated, nilpotent, non-nilpotent, and low-dimensional Leibniz algebras. In the present paper we complete the description for three-dimensional non-Lie left Leibniz algebras over arbitrary fields. We use a refined organization of the three-dimensional classification into sixteen isomorphism types, including parameterized families. Previously known cases are incorporated by reference, with changes of basis recorded when necessary. For the remaining cases we determine the derivations by direct coefficient calculations. Special attention is paid to characteristic two, where several derivation algebras increase in dimension. We also record simple decompositions of the resulting Lie algebras into natural ideals and subalgebras. A final table gives the derivation algebra of every type in the classification.

Automorphism groups are among the most natural structural invariants of an algebra, and their explicit determination is a basic problem in the structure theory of Leibniz algebras. In a series of earlier papers, automorphism groups were obtained for several classes of low-dimensional, one-generated, nilpotent, and non-nilpotent Leibniz algebras. In the present paper we complete this picture for three-dimensional non-Lie left Leibniz algebras over arbitrary fields. We use a refined organization of the three-dimensional classification into sixteen isomorphism types, including parameterized families. Previously known cases are not recomputed; instead, they are incorporated by reference, with changes of basis recorded when necessary. For the remaining types we determine the automorphism groups explicitly through a direct analysis of the automorphism conditions. Particular attention is paid to exceptional parameter values and to characteristic two. A final table summarizes the automorphism group of every type in the classification.

We give a unified study of local derivations on conformal Galilei Lie algebras in three natural settings: the algebra without central extension, the mass central extension, and the exotic central extension. We first determine the structural form of the derivation algebras and isolate the relevant outer derivations. For the mass extension, we complete the one-string rigidity at the two low parameters not covered by the previous one-dimensional treatment, using direct calculations, and combine this with new spatial compatibility arguments to obtain the result in arbitrary spatial dimension. The exotic extension is treated by a self-contained argument from its defining relations. For the algebra without central extension we resolve the remaining algebraic-reflexivity problem completely. If $d\ge2$, every local derivation is again a derivation. In spatial dimension $d=1$, the same conclusion holds when $2\ell$ is even and also for $\ell=\frac12$; however, for odd $2\ell\ge3$ there is an additional $(2\ell-1)$-dimensional space of pure local derivations. This gives a sharp contrast with the mass central extension, where the central Heisenberg pairing removes precisely this extra local freedom. We also determine the Lie algebra structure of the full local-derivation space: in the exceptional one-dimensional non-central case it is an explicit semidirect product with an additional abelian irreducible ideal.

Let \(A\) be a finite-dimensional quantum complete intersection over an arbitrary field. We prove that a finite-dimensional left \(A\)-module \(M\) is projective whenever \(\Ext_A^1(M,M)=\Ext_A^2(M,M)=0\), with no restriction on the multiplicative orders of the commutation parameters. Consequently, \(A\) satisfies the Auslander–Reiten conjecture and Tachikawa's second conjecture. The proof combines the gradability of rigid modules with successive graded twists and the self-extension criterion of Avramov and Buchweitz for commutative complete intersections. We derive a formula comparing ordinary self-extension groups under graded twisting, apply the resulting criterion to the Liu–Schulz algebras, and describe the successive twists explicitly in the four-generator case.

Over perfect fields of characteristic $p\ge5$, the unipotent radicals of the height-one special, Witt-Jacobson, and Hamiltonian automorphism groups in at least three, two, and four variables, respectively, attain the largest derived length allowed by their nilpotency classes. More strongly, every term of each derived series equals the corresponding power-of-two term of the lower central series. These equalities hold on finite-field points, with explicit Frattini quotients and lower $p$-central series. For the special family, a Cartier quotient measures the difference between the congruence and lower central filtrations. The Hamiltonian correction comes from flux and the socle line; the Witt-Jacobson family has none. We also identify the contact automorphism group scheme and, at arbitrary divided-power height, compute the Cartier-Verschiebung maps on restricted derivations and the Verschiebung filtration and distribution algebra of the Frobenius thickening of the special radical.

The shifted congruence series of finite special Nottingham jet groups is realized by an integral augmentation filtration. For volume-preserving formal substitutions in characteristic $p\ge5$ and $n\ge3$ variables, the filtration induced by the action on truncated power series has the congruence subgroups as its dimension subgroups. The canonical weighted augmentation filtration has the same dimension subgroups. The group symbols form a degree-truncated Lie algebra of divergence-free polynomial fields. Since brackets with quadratic fields give the subspace of exact fields in every coefficient degree at least three, the Lie indecomposables occur only in degrees $(p-1)(n-1)+pr$, with $r\ge0$, where they are determinant-weighted Frobenius twists of tensor products of symmetric powers with the defining module.

We study automorphic transcendence degree for quotients of multivariate skew polynomial rings over a division ring D. We identify this invariant with relative growth dimension, the degree of the relative Hilbert polynomial, and a coordinate dimension determined by leading monomials. We establish invariance under finite module extensions, integral extensions, and investigate the Gelfand-Kirillov dimension under a local finiteness condition. When the coefficient automorphisms are independent modulo inner automorphisms, we prove that a quotient is automorphically normalizable precisely when its automorphic transcendence degree equals the number of nonnilpotent coordinate variables, and classify all such normalization subrings.

We study finite left-nested Jordan products of idempotents in finite-dimensional simple Jordan algebras over a field $\mathbb{F}$ of characteristic different from $2$. Over an algebraically closed field, the structure theorem reduces the problem to the scalar algebra, spin factors, the full matrix Jordan algebras $M_n(\mathbb{F})^+$, the symmetric matrix algebras $H_n(\mathbb{F})$, the symplectic Hermitian algebras $H_{2n}(\mathbb{F},\mathrm{sp})$, and the split Albert algebra. We determine exactly which elements are representable: except in $H_2(\mathbb{F})$, these are all nonscalar elements together with $0$ and $1$, whereas in $H_2(\mathbb{F})$ the nonzero square-zero elements are also not representable. We also study the minimal word length of representable elements. For $M_n(\mathbb{F})^+$ over an arbitrary field, the maximal word length grows logarithmically in $n$, and for $n=2$ we determine the exact length of every nonscalar matrix. The same logarithmic growth holds for $H_{2n}(\mathbb{F},\mathrm{sp})$. The spin-factor and symmetric-matrix results hold over quadratically closed fields, whereas the full matrix, symplectic Hermitian, and split Albert results hold over arbitrary fields of characteristic different from $2$.

We construct a transferred $L_\infty$-structure on the shifted Chouhy–Solotar cochain complex of a finite-dimensional skew-gentle algebra and give explicit basis-level formulas for its binary operation $l_2$ in all shifted degrees. We further prove that the Hochschild dg Lie algebra of an $A_n$-type skew-gentle algebra is homotopy abelian, and hence $L_\infty$-formal, and contrast this with Wong's non-formality result for the dual numbers $\mathbb{K}[\theta]/\langle \theta^2 \rangle$, leading to a general formality classification problem.