Papers

In this paper we consider matrices whose entries are rational functions of several parameters over a Moore-Penrose field, that is, a field with an involutory automorphism over which every matrix has Moore-Penrose inverse. We prove that over any such field the Penrose conditions can be solved with linear algebra alone, even though they form a polynomial system of degree two. We then bound the degrees of the numerator and of the denominator of the pseudoinverse in terms of the degree of the entries and of the rank, and not of the dimensions of the matrix. Specialization is the thread: the closed form is computed once over the field of rational functions, and the question is at which parameter values it still returns the pseudoinverse of the specialized matrix. We determine the values where it does not, from the matrix alone and before computing the pseudoinverse. They are the zeros of a polynomial built from the maximal minors, and they are the values at which the rank decreases. This turns previous sufficient conditions into an exact characterization. The theory also gives symbolic algorithms for the real, the complex and the parametric case, which we implement in Maple and test on 972 timed replications. We also apply them to a Leontief economic model, where the excluded value is the point at which the economy stops being viable.

The alternating composition of $N = 2p$ weighted differential operators $w_j(x)\cdot\partial_x^{\,p}$ of strict order $p$ on the line $\mathbb{R} \ni x$ is again an operator of order $p$; its coefficient is the universal constant $c(p)$ times the Wronskian of the weights $w_1,\ldots,w_N$. Lie brackets of vector fields fix $c(p=1)=1$; we want to find $c(p \geqslant 2)$: e.g., $c(2) = 2$ or $c(3) = 90$. Direct symbolic expansion (over $|S_{2p}| =(2p)!$ permutations) fails for $p \geqslant 4$. Taking the monomials $w_j = x^{j-1}$ reduces the summation to the much smaller set $\Phi_p \subseteq S_{2p-1} \subsetneq S_{2p}$ of late-growing permutations. Expressing $c(p)$ as a signed sum of products of falling factorials, we implement and speed up the algorithm that gains all the integer values up to $c(18) = 4.881\ldots \cdot 10^{462}$. The resulting sequence is new, now registered as OEIS A392714; its (sub)leading-order growth rate is $\log c(p) \simeq 2p^2\log p -b p^2 + \overline{o}(p^2)$ for $p\gg 1$, with $b\geqslant 2.6744$.

For an Alexander quandle on an abelian group $G$ with automorphism $\phi$, we identify its augmentation powers and nonunital commutator subalgebra with explicit ideals in the ordinary group ring over any commutative unital coefficient ring. Each left-normed augmentation power is also the span of all products with the same number of factors, independently of parenthesization. Over $\mathbb{Z}$, we determine every successive augmentation quotient for Alexander quandles on nontrivial finite cyclic groups. For an even dihedral quandle of order $m \ge 4$, every quotient in degree $r \ge 2$ is $(\mathbb{Z}/(m/2)\mathbb{Z})^2$; this proves the proposed replacement for the disproved order-$m$ prediction. For the integral quandle ring of $\operatorname{Core}(\mathbb{Z})$, we describe all augmentation powers, determine the nonunital commutator subalgebra, and prove that the difference of the basis elements indexed by 1 and 0 does not lie in the square of the augmentation ideal. When $(1-\phi)G$ is a finite $p$-group, the integral filtration is separated, so every nonzero integral idempotent has augmentation one. Fourier analysis shows that finite odd-order commutative Alexander quandles have only zero and the basis elements as integral idempotents. Combined with the midpoint description of medial commutative quandles, this gives the same classification for every finite medial commutative quandle. Finally, the complex quandle algebra of the core of any nontrivial finite abelian group of odd order has right Peirce spectrum $\{0,1,-1\}$, proving the conjectured odd-order dihedral spectrum.

We determine the first extension groups between simple restricted modules for the restricted simple Lie algebras of Cartan type over an algebraically closed field of characteristic $p\ge5$, which answers the restricted case of a problem of Benkart and Feldvoss. Extensions from parabolically induced modules are computed by finite restricted-cohomology complexes of the grading parabolic, and for the exceptional modules we obtain closed formulas in all four families. Neither the trivial module nor any nontrivial exceptional simple has self-extensions. In characteristic seven we determine the quiver of $S(3;1)^{(1)}$, which has 49 vertices and 566 arrows.

For two bigraded spaces $k^{m||n}$ and $k^{n\wedge m}$ equipped with appropriate quadratic Poisson structures, Koszul duality identifies the mixed Poisson differential calculi and, in the unimodular case, their Poisson cohomologies as Batalin–Vilkovisky algebras. In the current paper we prove that Koszul duality induces an isomorphism of Batalin–Vilkovisky algebras between their Poisson cohomologies when the modular derivation of one algebra is diagonalizable, including non-unimodular cases.

Kubo-Ando means are a rich class of important binary operations on the set of positive elements of an operator algebra. The aim of this article is to understand the algebraic structure of these operations. We generalize a recent result of L. Molnár for $C^*$-algebras to the class of split $C^*$-by-nilpotent algebras by using the method of analytic local expansion of binary operations. Eventually, we prove that in any $C^*$-algebra $\mathcal{A}$, the identity $[x,[y,z]]=0$ implies commutativity.

For a finite ring $R$, we investigate the probability that the product of two randomly chosen elements in $R$ is nilpotent, which we call the nilpotent product probability (NPP) of $R$ and denote by $P_{nil}(R)$. We derive bounds for $P_{nil}(R)$ that depend on the structure of $R$, and prove that among all commutative rings with identity, the class of local rings with residue field $\mathbb{Z}_{2}$ attains the maximum value of NPP, which is equal to $3/4$. Further, we determine the set of all values of NPP for commutative rings with identity and classify all rings attaining those values. Finally, we investigate the relationship between NPP of $R$ and NPP of some subrings of the matrix ring over $R$.

Noncommutative polynomials, elements of the free algebra generated by several noncommuting variables $x_1,...,x_g,x_1^*,...,x_g^*$, are of importance in noncommutative algebra, free analysis and real algebraic geometry. We allow matrix coefficients and view these noncommutative polynomials as functions in several size-free matrix variables. The free locus of a noncommutative polynomial $f$ is the sequence of hypersurfaces $\mathscr{Z}_n(f)=\{X\in M_n(\mathbb{C})^g:\det f(X) = 0\}$. When $f$ is Hermitian, the positivity domain of $f$ is the sequence of semialgebraic sets $\mathcal{D}_n(f)=\{X\in M_n(\mathbb{C})^g:f(X)\text{ is positive semidefinite}\}$. We show for well-behaved cases that the Euclidean boundary of the positivity domain is Zariski dense in the free locus. This leads to a certificate for equality of positivity domains of coefficient-irreducible unsignatured atomic quadratics. Finally, we describe the geometric structure of positivity domains of hereditary quadratic polynomials through $\Gamma$-convexity.

Let $\Bbbk$ be an algebraically closed field of characteristic zero and let $f$ be a nonconstant polynomial in finitely many freely noncommuting variables. We prove the asymptotic Makar-Limanov rank conjecture, namely that the normalized rank of a value of $f$ can be made arbitrarily small by choosing matrices over $\Bbbk$ of finite size.

We study the dimension defect of finitely many subspaces over an arbitrary field. We derive a symmetric, nonrecursive Grassmann-type formula for the dimension of their sum. The formula expresses the total dimension loss through intersections of a distinguished subspace with partial sums of the remaining subspaces, with coefficients determined by the number of subspaces involved. We also show that the formula admits a Shapley-value interpretation for the associated representable polymatroid. Grouping the correction terms by the number of participating subspaces yields a nonnegative dimension-defect profile. We prove that this profile is monotone and identify its successive gaps with the discrete curvatures of the average-rank profile. These gaps give exact remainders in two-sided defect bounds. Equality in either bound holds precisely when the images of the subspaces in the quotient by their common intersection form an internal direct sum. We also give an exact geometric decomposition of the Kinser slack into nonnegative quotient dimensions and characterize equality. Averaging these slacks over permutations and contractions recovers every defect curvature except the final one; additional representability constraints remain in the individual ordered slacks. Weighted and dual formulas accompany the expansion, and entropy analogues express the defect levels and curvatures as averages of mutual and conditional mutual information, respectively.

In this article, we show that the characterization of reversible rings and various other rings related to them such as i-reversible rings, semi-commutative rings and i-domains in terms of properties related to idempotent elements known in the literature can be generalized to exactly the same properties for quasi-idempotent elements.

We study finite Laver algebras, the generalizations of Laver tables on sets of cardinality not of the form $2^n$. We characterize the conditions under which projections and embeddings between Laver algebras exist and show that there are $2^{\aleph_0}$ pairwise non-isomorphic direct limits of finite Laver algebras.

We classify, up to graded isomorphism, graded bimodules over split group algebras of finite abelian subgroups of an arbitrary group. The classification is governed by double cosets, stabilizer characters, and coefficient modules. Over a field, we determine the essential image of the induction functor on Yetter-Drinfel'd modules, giving a negative answer to a question of Dos Santos and Yasumura in the nonabelian case. We also compute the Peirce components and characterize multiplicity-free bimodules through extensions of stabilizer characters.

We extend the theory of Baer, Rickart, and their local, graded, and $*$-analogues for Steinberg algebras. First, we prove that positive definiteness passes from the coefficient field to the Steinberg algebra, and that this, together with graded von Neumann regularity, characterized by Steinberg and van Wyk, yields graded locally Rickart $*$-Steinberg algebras. Then we show that local Baer behavior forces every compact open subset of the unit space to be extremally disconnected. Consequently, when compact open subspaces of the unit space are metrizable, local Baer behavior forces the groupoid to be discrete. In the discrete case, we use the orbit-by-orbit decomposition into finitary matrix algebras over isotropy group algebras to convert annihilator conditions into explicit matrix conditions. For groupoids whose isotropy is trivial or infinite cyclic, these ingredients yield complete local and unital characterizations. In particular, local Baer and graded local Baer conditions are equivalent to discreteness, while the ungraded local Baer $*$-condition additionally requires that every orbit with nontrivial isotropy be a singleton. We then apply the theory to boundary path groupoids. Using the Steinberg algebra model, we obtain the Rickart, Baer, and Baer $*$ characterizations of Leavitt path algebras, including the local and graded variants.

For every $N\ge36$, we realise any prescribed monic irreducible binary polynomial of degree $N$ as the characteristic polynomial of a linear hybrid cellular automaton on a near-square region of exactly $N$ cells. The transition matrix is Moore-local with a null boundary and has at most $3N-1$ directed nonself dependencies. The dependency graph retains a bidirectional Hamilton path; its indegree, outdegree and underlying undirected degree are at most six. The underlying graph contains an explicit square grid minor of side proportional to $\sqrt N$. The deterministic synthesis takes $O(N^3)$ bit operations. The construction combines a local similarity transformation with a transport potential and joint routing across consecutive row gaps. Exact verification of finite certificates, followed by induction, proves the required routing inequalities for every admissible width. We also give an entirely analytic construction with fewer than $7N/2$ dependencies and prove a lower bound of $5N/2-O(\sqrt N)$ for the retained path and full rectangular grid minor. This lower bound is sharp when the characteristic polynomial is unrestricted.

We prove the finiteness of the genus of quaternion division algebras over some formally real fields. In particular, if $F$ is a formally real field and the second power of the fundamental ideal $I^2(F)$ is torsion-free, then the genus of any quaternion division $F$-algebra is trivial. For instance, this immediately implies that the genus vanishes for any quaternion division algebra over a formally real pythagorean field. The proofs are based on methods from the theory of quadratic forms.

In 1999, A. Smoktunowicz solved a long-standing problem and constructed a simple nil ring. We present a substantial simplification of this construction found by Aristotle (Harmonic) with the assistance of ChatGPT.

We study the realization problem for point schemes of skew polynomial algebras. We approach this problem combinatorially by translating it into the study of certain $3$-uniform hypergraphs, which we call point hypergraphs. We first give a homological criterion characterizing point hypergraphs. Using this criterion, we study hypergraphs obtained by deleting triangles from triangulations of connected orientable closed surfaces. Deleting exactly one triangle always gives a non-point hypergraph. When the triangulation has no separating nonfacial $3$-cycles, deleting either no triangles or at least two triangles gives a point hypergraph, and the one-triangle deletions are minimal non-point hypergraphs with respect to taking induced sub-hypergraphs. It follows that point hypergraphs cannot be characterized by finitely many forbidden induced sub-hypergraphs. Next, for each point hypergraph, we construct an affine moduli variety of skew polynomial algebras realizing it, and determine its dimension. For point hypergraphs arising from the above surface construction, we obtain an explicit dimension formula in terms of the number of vertices, the Euler characteristic, and the number of deleted triangles. Finally, we characterize point hypergraphs on six vertices in terms of a four-vertex local condition together with a single exceptional obstruction.

We give a short and simple proof that an anisotropic generalized quadratic form over a quaternion division algebra with involution of the first kind, in arbitrary characteristic, remains anisotropic after scalar extension to any odd-degree field extension.

Abel's classical theorem shows that a continuously differentiable one-dimensional associative law with a two-sided unit is locally equivalent to addition under a change of coordinate. We prove an $n$-valued analogue in the algebraic and formal settings. Over a field of characteristic zero, every associative algebraic $n$-valued monoid on $\mathbb P^1$ with a strong two-sided unit is commutative. Over a reduced commutative $\mathbb Q$-algebra, any one-dimensional formal $n$-valued group is commutative. We then classify $n$-valued formal groups over arbitrary $\mathbb Q$-algebras that are integral domains: for any $n$-valued formal group there is a unique divisor $m$ of $n$ such that the law is strongly isomorphic to the $n/m$-fold diagonal of the cyclic elementary $m$-valued law $\mathbb G_m$. This proves Kholodov's classification conjecture over integral domain $\mathbb Q$-algebras.

We study formal matrix rings with the involution property, that is, rings in which every element can be represented as the sum of an invertible element and an involution. After recalling the construction of formal matrix rings via Morita contexts, we describe involutions in triangular formal matrix rings and establish a criterion for upper triangular rings of order two. We then prove that a formal matrix ring of arbitrary finite order has the involution property whenever all of its diagonal rings have this property. Several examples and counterexamples are included to illustrate the obtained results.

We introduce the notion of a Cartan inverse semigroup and an (algebraic) semi-Cartan subalgebra in an $R$-algebra A, providing a purely algebraic analogue of Cartan semigroups and semi-Cartan subalgebras in a C*-algebra. This extends the theory of quasi-Cartan subalgebras, shifting the focus from designated subalgebras to designated subsemigroups. Our main result characterises twisted Steinberg algebras in terms of the existence of a Cartan inverse semigroup: we show that an $R$-algebra admits a Cartan inverse semigroup if and only if it is isomorphic to a twisted Steinberg algebra. Moreover, when our construction is applied to a twisted Steinberg algebra over an ample Hausdorff groupoid, the constructed twist is naturally isomorphic to the original one.

Let $\mathcal O$ be an octonion algebra over a field $F$ of characteristic different from $2$, and let $A,B\in\mathcal O$, not both zero. We study the image of the generalized commutator $\Phi(x,y)=A(xy)-B(yx)$ on $\mathcal O$. We prove that $\Phi$ is surjective if and only if either one of $A$ and $B$ is invertible and $A\neq B$, or $A$ and $B$ are nonzero zero divisors and $A-B$ is invertible. In the remaining cases, the image of $\Phi$ is a hyperplane or a four-dimensional maximal totally isotropic subspace of $\mathcal O$, except when $A$ and $B$ are linearly independent zero divisors and $A-B$ is not invertible, in which case the image spans a six-dimensional subspace. Along the way, we show that every octonion of trace zero is a commutator. We also prove that the image of $Axy-Byx$ on any quaternion algebra over $F$ is a vector subspace, extending a result of Panja S., Saini P. and Singh A. (Images of polynomial maps with constants, Mathematika 71 (2025), no. 3, Paper No. e70031) for $2\times 2$ matrices over algebraically closed fields, and we give examples of other generalized polynomials whose images are not vector subspaces.

In 1978, Amitsur and Small asked whether, for every division ring $D$, maximal left ideals of $D[x_1,\ldots,x_n]$ contract to maximal left ideals in smaller polynomial subrings. Chapman and the author showed that the answer is negative in general and called $D$ an Amitsur-Small ring when this contraction property always holds. Earlier work showed that Hamilton's real quaternion algebra is Amitsur-Small, that division algebras of degree three are not Amitsur-Small, and that degree-two examples are restricted to a specific quaternionic form; subsequent work excluded cyclic division algebras of odd prime degree. We give a complete resolution in the centrally finite case. If $D$ has finite dimension over its center $F$, then $D$ is an Amitsur-Small ring if and only if either $D=F$, or $F$ is real closed and $D$ is the Hamilton quaternion algebra $(-1,-1)_F$. In every other noncommutative centrally finite case, failure already occurs in two variables: there is a maximal left ideal $M\subseteq D[x,y]$ such that $M\cap D[x]$ is not maximal in $D[x]$.

In this work we provide the complete classification of all non-zero finite-dimensional complex Lie subalgebras of the universal enveloping algebra of the special linear Lie algebra $\mathfrak{sl}(2,\mathbb{C})$. That is, we show that every such nilpotent Lie subalgebra is abelian and every non-solvable Lie subalgebra is reductive, with semisimple component isomorphic to $\mathfrak{sl}(2,\mathbb{C})$ itself. Every non-nilpotent solvable Lie subalgebra is the direct sum of an abelian center and a member of the family of solvable Lie algebras $\mathfrak{r}(\boldsymbol{d},\boldsymbol{m})$, parameterized by a vector $\boldsymbol{d}$ whose entries are strictly increasing positive integers that have greatest common divisor one together with a positive integer vector $\boldsymbol{m}$ of the same length recording their multiplicities. Moreover, we discuss the corresponding classification in a real skew-hermitian setting and its relation to the realizations in the single- and two-mode Weyl algebras.

Let X and Y be finite ordered sets and let $X^Y$ denote the ordered set of order-preserving maps from $Y$ to $X$. Let $t$ be a~term formed from a~single variable by exponentiation. We prove that, for every term $t$ with at most twelve variable occurrences and arbitrary finite ordered sets $A,B$, $t(A)\cong t(B)$ implies that $A\cong B$. General constructions and reconstruction principles are developed first; the second part gives the occurrence-by-occurrence proofs. At twelve occurrences the proof covers 4,766 interchange normal forms, representing all 58,786 binary parenthesizations, including 40 exceptional forms. Two additional unbounded reconstruction families, complete finite audit data, and 457 exact height-comparison certificates are included.

The theory of cluster algebras is closely connected to the theory of total positivity; indeed, the desire to better understand total positivity was one of the main motivations for Fomin and Zelevinsky's introduction of cluster algebras [arXiv:math/0104151]. In particular, any cluster variety whose coordinate ring has a cluster structure has a natural notion of positive part: the subset of the variety where all cluster variables are positive. In this paper, we explain that there are other signed cells contained in cluster varieties that are equally natural from a cluster-theoretic point of view. These come from signed seeds, which can be thought of as a $\mathbb{Z}/2\mathbb{Z}$-grading on cluster variables, and which were introduced in [arXiv:2310.17727] in the context of the amplituhedron. More generally, given any abelian group $G$, we introduce the notion of a $G$-graded seed for a cluster algebra, which is a way of assigning elements of $G$ to each cluster variable which is compatible with the cluster structure. When $G$ is the multiplicative group $\{-1, 1\}$, this recovers the above notion of signed seed; when $G = \mathbb{C}^*$, this recovers the notion of cluster automorphism group [GSV10] or cluster dilation group [arXiv:2603.17890]; and when $G = \mathbb{Z}^d$, this recovers the notion of graded cluster algebra studied by Grabowski-Launois [arXiv:1301.2133], Grabowski [arXiv:1309.6170] and Gekhtman-Shapiro-Vainshtein [GSV10, Section 5.2] (which had previously appeared in special cases in work of Fomin-Zelevinsky [arXiv:math/0602259]). The examples we study include the space of square matrices, symmetric matrices, skew-symmetric matrices, positroid varieties, and amplituhedron tiles. We also connect this notion to tropical mutation when $G = \mathbb{R}$ or $\mathbb{Z}$.

An algebra with zero is an algebra that contains an element 0, called zero, such that all fundamental operations have the property that the result is zero whenever at least one of the arguments takes the value 0. We introduce the notion of the higher commutator of ideals and Rees congruences for such algebras. It is shown that the commutator of Rees congruences is equal to the Rees congruence of the commutator of the corresponding ideals. We apply this result to provide characterisations of n-supernilpotency and n-nilpotency in algebras with zero, as well as n-solvability in groupoids with zero.

An algebra is called binary Zinbiel if every subalgebra generated by at most two elements is a Zinbiel algebra. Over a field of characteristic zero, we prove that every finite-dimensional binary Zinbiel algebra is nilpotent. We also show that a subvariety of binary Zinbiel algebras is nil if and only if it does not contain the variety of all Zinbiel algebras. We further prove an analogue of Itô's theorem for binary Zinbiel algebras.

We study derivations and automorphisms of arbitrary non-abelian extensions of $n$-Lie algebras through their full crossed-product data. Their relative center is a canonical coefficient module, and a third cohomology class obstructs completion of data satisfying the mixed Filippov identities. When this class vanishes, a relative second cohomology space classifies the remaining extension data. We prove a non-abelian good criterion, construct the Lie algebra and group of compatible pairs, and obtain derivation and automorphism Wells sequences with explicit lifting equations. The derivation obstruction is a Lie algebra $1$-cocycle; the additive automorphism obstruction is a crossed homomorphism. We distinguish the ordinary center from the relative center, and outer derivations of the kernel from the mixed action required. Sparse crossed products are treated, including the representation class of their second mixed operation and the additional equations governing changes of sparse sections.