Papers

A pseudo-Euclidean Malcev algebra is a Malcev algebra equipped with a non-degenerate symmetric bilinear form. In this paper, we introduce a curvature operator for pseudo-Euclidean Malcev algebras, generalizing the notion of curvature for pseudo-Euclidean Lie algebras. We define flat pseudo-Euclidean Malcev algebras and show that every flat Euclidean solvable Malcev algebra which is also a Lie algebra remains flat in the classical Lie algebra curvature sense. Furthermore, we develop the flat double extension construction for flat pseudo-Euclidean Malcev algebras and prove that every flat Lorentzian Malcev algebra with a degenerate center can be obtained via the flat double extension of a flat Euclidean Malcev algebra. Moreover, we demonstrate that all flat Lorentzian nilpotent Malcev algebras arise from the flat double extension of a Euclidean abelian Lie algebras. Finally, we establish that any flat Lorentzian nilpotent Malcev algebra is necessarily a Lie algebra and is flat in the classical Lie algebra curvature sense.

We investigate the structure of Poisson algebras of dimensions at most three over an arbitrary field. Our approach is based on the internal structure of the associated commutative associative and Lie algebras, with particular emphasis on the associative square $P^2$, the derived Lie algebra $[P,P]$, the Lie center, the associative annihilator, idempotents, ideals and decomposability. We obtain a complete classification in dimensions one and two and give a structural classification in dimension three. In dimension two, we prove that the associative and Lie multiplications cannot be simultaneously non-zero. In dimension three, the classification is organized according to the dimension and position of the derived Lie algebra and, in the case of trivial Lie multiplication, according to the dimension of $P^2$. The arbitrary-field setting leads to phenomena which do not occur over the complex field. In particular, quadratic and cubic field extensions appear naturally in the classification, some families depend on equivalence classes of symmetric bilinear forms and on the structure of three-dimensional Lie algebras over the ground field, and characteristic $2$ gives an additional family of Poisson algebras with both multiplications non-zero. Over the complex field, the resulting classification specializes, up to changes of basis and notation, to the known classifications in dimensions at most three.

We study Poisson algebras in which every Poisson subalgebra is a Poisson ideal, called Dedekind Poisson algebras, over arbitrary fields and without any finite-dimensionality assumption. In characteristic different from 2, combining the two operations by $x*y=xy+[x,y]$ connects the problem with Outcalt's classical classification of power-associative H-algebras and yields the same structural type. We obtain a complete classification by a direct two-operation argument valid in every characteristic. This approach also shows that characteristic 2 retains additional Poisson information which cannot be recovered from the single product $*$.

For fixed nonnegative integers $m_1,m_2$, let $A_n=(a_{j-i})_{i,j=1}^n$ be the leading $n\times n$ section of a Toeplitz matrix with lower and upper semibandwidths $m_1$ and $m_2$. We give two constructive Laplace-expansion methods for scalar recurrences of $\det A_n$ and $\perm A_n$. The increasing-rows method eliminates a fixed family of boundary cofactors and gives recurrence order at most $d=\binom{m_1+m_2}{m_1}$ for both sequences. The row-column method closes normalized boundary minors recursively and packages them in a sparse transfer matrix. Its reachable states are classified exactly: level $j$ is indexed by a pair of $j$-subsets of $[m_1]$ and $[m_2]$. Hence the transfer dimension is $d$, and we obtain an explicit formula for the number of nonzero transitions. For determinants, the complementary cofactors of the increasing-rows construction are coordinates of the classical compound companion representation. The independently constructed row-column transfer has the Widom characteristic polynomial and is generically similar to the compound transfer. Thus the order $d$ recurrence is generically minimal for the unrestricted fixed-band determinant family. For permanents the same state graph gives the binomial upper bound, without a general minimality claim. The pentadiagonal case recovers Sweet's order-six determinant recurrence and its permanent analogue, while the one-superdiagonal family gives closed scalar recurrences and rational generating functions. Position-dependent band weights preserve the finite state graph but replace the constant transfer by a cocycle. For cyclic determinants, Fourier diagonalization produces all subset products of the symbol roots and a generically minimal annihilator of degree $2^{m_1+m_2}$, corresponding to the passage from one exterior degree to the full exterior algebra.

The AES S-box is constructed from finite field inversion followed by a fixed affine transformation. Since inversion possesses intrinsic Frobenius symmetries among its coordinate realizations, we study how these basis symmetries are altered by outer affine transformations. We first develop a deterministic rigidity criterion for transformed inversion and apply it to the AES S-box. This shows that the linear part of the AES S-box affine transformation alone makes the transformed inversion map basis rigid. We then investigate the corresponding generic problem when the outer invertible linear transformation varies. The existence of a nontrivial linear stabilizer is reduced to a conjugacy problem for semilinear candidates arising from two sided linear equivalences of inversion, which we characterize in terms of relative norms and Frobenius orbits. We also determine the dimensions of the associated centralizer algebras exactly. These structural results imply that, for a uniformly chosen outer linear transformation, the probability that the linear stabilizer is nontrivial is bounded by $2^{-\Omega(n^2)}$, with sharper finite dimensional bounds obtained from the exact conjugacy condition. Computational experiments independently verify the AES rigidity result, the conjugacy and centralizer formulas, and the finite dimensional estimates in small dimensions.

We study relative global dimensions of extensions $B \subseteq A$ of Artin algebras over a perfect field. A controllable extension is introduced as one for which the relative global dimension is determined by the ordinary global dimension of the quotient $A/AJ(B)A$. General inequalities and sufficient conditions for controllability are established, including the case in which $J(B)$ is a two-sided ideal of $A$. The interaction between relative global dimensions and relative projective dimensions of the regular bimodule is then studied. The class of homologically controllable extensions is introduced, and it is shown to be closed under tensor products. Consequently, for homologically controllable extensions $B \subseteq A$ and $D \subseteq C$, we obtain \[\operatorname{gldim}(A \otimes C, B \otimes D)=\operatorname{gldim}(A,B) + \operatorname{gldim}(C,D).\] The proof relies on a relative Künneth-type result for tensor products of complexes, which may be of independent interest. Finally, controllable extensions with prescribed relative global dimension are constructed, along with families of non-controllable extensions for which the relative global dimension differs arbitrarily from the global dimension of $A/AJ(B)A$.

In this work, we complete the description of the one-dimensional partial actions of 8-dimensional Hopf algebras by computing the remaining cases: the Kac-Paljutkin algebra $\mathcal{A}$ and the unique non-semisimple non-pointed Hopf algebra $\mathcal{K}$. We prove that all these partial actions are symmetric, study their associated partial smash products and we determine all their partial coactions of dimension one. Beyond the 8-dimensional setting, we investigate which right coideal subalgebras of a Hopf algebra $H$ can be realized as partial smash products $\underline{ \Bbbk \# H}$ over the base field. In particular, we show that every right coideal subalgebra of a finite-dimensional cosemisimple Hopf algebra arises in this way.

Let $R$ be a unital ring equipped with a Sylvester matrix rank function $\rk$. A harmonic function $\alpha$ on a Bratteli diagram $B$ defines a weighted matrix rank on the associated algebraic direct limit $A(B,R)$. We prove that, if $\alpha$ is extreme and the total weight of blocks of any fixed bounded size tends to zero, then the rank completion of $A(B,R)$ is isomorphic to $\mathcal M_{R,\rk}$, the rank completion of the direct system $\Mat_{2^k}(R)$ with connecting maps $x\mapsto\diag(x,x)$ and normalized ranks $2^{-k}\rk$. The isomorphism preserves the unital $R$-algebra structure and the ranks on all rectangular matrices. The coefficient ring need not be regular, and the specified rank need not be induced from a regular ring. We recover factor-sequence uniqueness and construct corners of every prescribed rank in $(0,1]$ that are isomorphic to $\mathcal M_{R,\rk}$ with their normalized ranks. Examples show that the coefficient rank can affect the isomorphism type and that the completion can be non-regular and non-simple. For complex coefficients, the trace determined by $\alpha$ gives a rank completion of the associated AF $C^*$-algebra canonically isomorphic to the affiliated-operator ring of its GNS closure. The rank completion of the algebraic direct limit can be a proper subring of this ring.

We prove that every increasing, polynomially bounded function is realizable, up to bi-Lipschitz equivalence, as the growth function of some finitely generated Lie algebra. This does not extend to general increasing subexponential functions. We further prove that the classes of growth functions of Jordan algebras and associative algebras coincide, answering in the affirmative a question of Martínez and Zelmanov.

This paper investigates reversed product properties of two elements in rings. Motivated by Cline's formula, we characterize reversible and $\ast$-reversible rings in terms of group invertible elements, EP elements, and the transfer behaviour of generalized inverses for reversed products. We prove that a unital ring $R$ is reversible if and only if $ab\in R^{\sharp}$ yields $ba\in R^{\sharp}$. For an involutive ring $R$, $R$ is $\ast$-reversible precisely whenever $ab\in R^{\mathrm{EP}}$ implies $b^{\ast}a\in R^{\mathrm{EP}}$. Several counterexamples are constructed to differentiate these ring classes, and the mutual inclusion relations among them are also discussed.

Given two finite essential adjacency matrices $A$ and $B$, Hazrat's graded classification conjectures posit that an order preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism of $K_0$ groups implies graded Morita equivalence of the Leavitt path algebras of $A$ and $B$, while the pointed version predicts a graded isomorphism of the Leavitt path algebras when the $K_0$ group isomorphism additionally preserves the class of the regular module. For any field $k$, we show that the Leavitt path algebras over $k$ of $A$ and $B$ are graded Morita equivalent if and only if $A$ and $B$ are strong shift equivalent. By appealing to counterexamples of Kim and Roush from symbolic dynamics, this shows that Hazrat's graded classification conjectures are false.

Direct evaluation of the defining product in the real Cayley–Dickson algebra $A_n$, of dimension $N=2^n$, has quadratic arithmetic complexity. This paper gives a uniform algorithm for multiplication using $O(N\log N)$ real arithmetic operations and $O(N)$ auxiliary storage. The algorithm reduces multiplication to the alternating product on the imaginary subspace, then evaluates that product by a two-call recursion over one fixed quadratic coefficient extension. For $n\ge1$, the resulting bilinear algorithm uses at most $(9n-15)2^{n-1}+10$ input-dependent real multiplications, and for $n\ge3$, the specified arithmetic schedule uses $(34n-83)2^{n-1}+50$ real operations in total. Under this counting convention, the quasilinear schedule uses fewer operations than direct multiplication for $N\ge16$ and than the uniform Cariow–Cariowa method for $N\ge32$. The algorithm is implemented in the MIT-licensed C11 library fastCD, with a NumPy-backed Python interface, and its results are checked against an independent implementation of the defining recursion. In single-core benchmarks against direct multiplication and the uniform Cariow–Cariowa method, the quasilinear implementation had the lowest mean time of the three at every tested dimension $N\ge32$, for both single and batched products, and was roughly $16$ times faster than direct multiplication at $N=1024$.

We consider the problem of computing rational invariants of nilpotent Lie algebras. We compare two methods that are commonly used for this task: the method of integral curves and the Dixmier map. Given a derivation of a rational function field with polynomial coefficients, we formulate a condition under which the kernel can be recovered from a family of rational integral curves, and we show that triangular derivations satisfy this hypothesis. This yields an explicit description of the kernel as a purely transcendental extension and produces algebraically independent generators. We also show that, in the triangular case, the resulting generators agree with those obtained from the Dixmier map via a local slice. A careful analysis of the generating set obtained from this method leads to an algorithm for computing generators of the rational invariant field of a nilpotent Lie algebra. An implementation of the methods is available in the SageMath system.

We prove that $M_n(\mathbb{C})$ contains no abelian Kadison–Singer algebra for any $n \geq 3$.

A completion theory for hyperbolic distance data is developed at the interface of matrix analysis, graph theory, and hyperbolic geometry. Krein's characterization of the metric space embeddability in Lobachevsky space leads to a natural anchoring procedure that transforms the indefinite data into a positive semidefinite kernel. In analogy with positive semidefinite and Euclidean distance matrix completion, chordality of the specification graph is shown to be the necessary and sufficient condition for local Lorentz-Gram data to admit global completion. Existence is complemented by explicit constructions. For trees, we obtain geodesic-rectification and product-distance completions; for chordal graphs, the latter extends to matrix-valued transfers along clique-trees. The resulting canonical completion is characterized by sparsity of its inverse and by a maximum-absolute-determinant principle. Its metric distortion exhibits a sharp dichotomy governed by clique separator size. Applications to exact recovery from sparse hyperbolic measurements and to hierarchical and phylogenetic data are developed.

We propose to define amenability of a Lie algebra by the existence of almost-invariant finite-dimensional subcoalgebras of its universal enveloping algebra. More generally, a module coalgebra of a cocommutative Hopf algebra is amenable if it admits almost-invariant finite-dimensional subcoalgebras. We prove a coalgebraic rounding theorem: almost-invariant finite-dimensional subspaces can be replaced, without loss in the Følner constant, by almost-invariant finite-dimensional subcoalgebras. For Lie algebras this implies that our definition is equivalent to Elek's amenability of the left regular module of the universal enveloping algebra seen merely as an associative algebra. For groups this recovers the result that a group is amenable if and only if its group ring is algebraically amenable. It furthermore shows that, for every amenable group, all its nonzero modules are amenable, thus proving an assertion by Gromov. We prove that amenable Hopf algebras are closed under taking subalgebras, quotients, cleft extensions, and directed unions, and that every Hopf algebra locally of subexponential growth is amenable. We give examples of amenable Lie algebras which are not elementarily amenable. Finally, we show that amenability passes to the associated graded Hopf-module coalgebra.

We give a simple proof for the existence of weak central polynomials for the algebra of multiplications of a finite-dimensional simple (non-associative) algebra. As an example we present explicit weak central polynomials in the cases of the three-dimensional simple Lie algebra and the Jordan algebra of the two-dimensional vector space with non-degenerate symmetric bilinear form.

Let $\mathbb{F}_{q^n}$ be a finite field where $q=p^l$ ($p$ is a prime) and a Dickson nearfield $R$ associated to the field. In this note we investigate the basic properties of the generalized distributive set $D(a,b)$ where $a,b\in R$. The set is well-known when $n=1, 2$. For $n>2$, we characterize when $D(a,b)$ is a subfield of $\mathbb{F}_{q^n}$. Furthermore we find the unique largest subfield of $\mathbb{F}_{q^n}$ contained in $D(a,b)$. We also provide a condition to guarantee that $D(a,b)$ is not a subfield of $\mathbb{F}_{q^n}$. We end the note by posing the question of when $D(a,b)$, as a space over $\mathbb{F}_p$, has a primitive normal basis.

We study Ringel's brick chain filtrations over finite-dimensional algebras: filtrations whose factors are filtered by copies of individual bricks, ordered so that morphisms from earlier bricks to later ones vanish. First, using submodule varieties, we prove that the largest submodule of a fixed module belonging to a torsion class takes only finitely many values as the class varies, answering Pavón's question. We deduce finiteness of brick chain filtrations and the bound $2^{d^2}$ for modules of dimension $d$. For $\tau$-tilting finite algebras, we bound their number by the multinomial coefficient determined by simple composition multiplicities. Finally, we construct families of bricks over the three-arrow Kronecker algebra whose filtration counts grow exponentially in the square of composition length. We prove this growth using a Littlewood–Richardson formula for submodule counts and the hook-length formula. In particular, the counts eventually exceed the factorial of composition length. These results answer Ringel's two questions.

Let $p$ be a prime and let $A,B$ be nonempty subsets of the cyclic group $\mathbb{Z}_p$ with $|A|\neq |B|$. The Alon–Nathanson–Ruzsa theorem gives the lower bound $|A\rplus B|\ge \min\{p,\,|A|+|B|-2\},$ where $A\rplus B=\{a+b:a\in A,\ b\in B,\ a\neq b\}$ is the restricted sumset. The inverse problem of characterizing all critical pairs $(A,B)$ attaining equality was posed by Alon, Nathanson, and Ruzsa in 1996 and remains open. Recently, Liu and Qian solved the inverse problem under the assumption that at least one of the sets is an arithmetic progression, and proposed a conjecture for the general case. The main purpose of this paper is to show that this conjecture fails in the boundary case $|A|+|B|=p$. More precisely, we construct infinite families of non-arithmetic critical pairs $(A,B)$ with $|A|+|B|=p$ and $|A\rplus B|=p-2$ for every prime $p\ge 11$. These families show that the boundary case is fundamentally different from the non-boundary case. Motivated by this, we formulate and prove a refined inverse theorem under the natural hypothesis $|A|+|B|\le p-1$, which excludes the boundary.

Motivated by lifting problems for principal bundles, we study the following algebraic problem. Let $\smash{\widehat{G}}$ be a group with identity element $e$, let $G \leq \smash{\widehat{G}}$, and let $S$ be a strongly $G$-graded unital ring with principal component $R:=S_e$. We ask whether the given grading extends to a strong $\smash{\widehat{G}}$-grading without altering its $G$-homogeneous components. For every extension $\smash{\widehat{p}}$ of the Picard homomorphism of $S$, we construct a characteristic class in the relative third cohomology group \[ H^3_{\smash{\widehat{p}}}(\smash{\widehat{G}},G;\operatorname{U}(Z(R))), \] where $Z(R)$ denotes the center of $R$ and $\operatorname{U}(Z(R))$ its group of units. The~vanishing of this class is equivalent to the existence of a lift. When it vanishes, the relative second cohomology group \[ H^2_{\smash{\widehat{p}}}(\smash{\widehat{G}},G;\operatorname{U}(Z(R))) \] acts simply transitively on the equivalence classes of lifts with Picard homomorphism $\smash{\widehat{p}}$. We also prove that an $H$-grading is strong if and only if both its restriction to a normal subgroup $N \trianglelefteq H$ and its induced $H/N$-grading are strong. We illustrate the theory through a range of examples.

We give a counterexample to a weak majorization problem, proposed by Z. Huang (Linear Algebra Appl., 434 (2) (2011) 457–462), for singular values of Hadamard products of nonnegative matrices. The problem asks whether, for all nonnegative matrices $A$ and $B$, \begin{equation*} \bigl\{s_j^2(A\Had B)\bigr\} \prec_w \bigl\{s_j(A\Had A)s_j(B\Had B)\bigr\} \end{equation*} holds, where $\Had$ denotes the Hadamard product, $\prec_w$ means weak majorization, and $s_j(\cdot)$ is the $j$th largest singular value of a matrix. To answer this problem, we construct two $3\times3$ entrywise positive symmetric matrices. We give rigorous upper and lower bounds for the relevant singular-value sums by Sturm's theorem, and show that the inequality fails for the partial sum with $k=2$.

The periodicity conjecture asks whether a finite-dimensional algebra is periodic whenever all its simple modules are periodic. We construct a $36$-dimensional counterexample whose simple modules have period four. The algebra has a $2$-dimensional nonperiodic module, which shows that the algebra itself is not periodic.

Existing classifications of real plane cubic curves rely on sophisticated tools and require a lengthy exposition. This paper provides a concise yet elementary treatment of this classical topic. Using the centers of polynomials, we establish a correspondence between the algebraic structure of these centers and geometric properties of binary cubic polynomials, which yields a novel approach to Newton's canonical forms.

We compute quasi-derivations of Block Lie algebras $\mathcal{B}(q)$. In particular, we prove that $\mathcal{B}(q)$ has various nontrivial quasi-derivations, i.e., quasi-derivations different from derivations and $\delta$-derivations. As corollaries, we obtain a complete description of $\delta$-derivations, $\delta$-biderivations, and transposed $\delta$-Poisson structures on $\mathcal{B}(q)$ for $\delta\notin\{0;1\}$. In particular, for $\delta\notin\{0;1\}$, $\mathcal{B}(q)$ up to isomorphism admits only one nontrivial transposed $\delta$-Poisson structure if $q\in\mathbb Z$, and $\mathcal{B}(q)$ does not admit nontrivial transposed $\delta$-Poisson structures if $q\notin\mathbb Z$.

We provide a complete classification of Nijenhuis operators on three-dimensional multiplicative simple BiHom-Lie algebras over the complex field C. Based on the classification of such algebras by Saadaoui into three families L1, L2, and L3, we derive the complete system of equations arising from the equivariance conditions and the Nijenhuis identity. We provide explicit matrix representations of all solutions, including all degenerate parameter cases. We construct the BiHom-Frölicher-Nijenhuis bracket on the cochain complex of a BiHom-Lie algebra, showing that Maurer-Cartan elements of this graded Lie algebra are precisely Nijenhuis operators. This enables us to define the cohomology of Nijenhuis operators and study formal deformations. We introduce BiHom-NS-Lie algebras as the algebraic structure underlying Nijenhuis operators. We provide explicit deformed BiHom-Lie algebra structures and prove the non-isomorphism of different families. We also study the moduli space of Nijenhuis operators and provide computational verification of all results.

An infinitely divisible nonnegative matrix is an entry-wise nonnegative matrix that admits an entry-wise nonnegative $m$th root, with respect to usual matrix multiplication, for every positive integer $m$; it is called strongly infinitely divisible when it is, in addition, invertible. The study of such matrices has its origins in the theory of infinitely divisible probability distributions and the embedding problem for Markov matrices and is closely connected with continuous one-parameter semigroups; in the invertible case, this connection admits a natural description in terms of the exponential map. In this paper, we characterize the bijective linear maps on $M_n(\mathbb{R})$ that preserve strongly infinitely divisible matrices and infinitely divisible nonnegative matrices. In the former case, we combine the Inverse Function Theorem with Zariski-density techniques, using the Zariski-density approach recently developed in linear preserver theory by Fallat and Mondal [Proc. Amer. Math. Soc., 2026]. In the latter case, we first show that preserving infinite divisibility forces preservation of the cone of entry-wise nonnegative matrices and then exploit the additional structure afforded by infinite divisibility to complete the characterization.

We study direct products of monoids with zero and give a criterion under which their factors can be recovered from the multiplicative structure alone. While classical decomposition theory encodes direct products through factor congruences, central elements, and refinement properties, we give a concrete multiplicative reconstruction mechanism. If the factors have no nontrivial complemented central idempotents, then the coordinate idempotents are precisely the atoms and coatoms of the complemented-central-idempotent poset, and their multiplicative stabilizers are precisely the coordinate factors and cofactors. It follows that every isomorphism between such products is monomial, yielding the corresponding wreath-product description of automorphism groups. More generally, every product decomposition is obtained by grouping the original factors; in particular, strict refinement follows. We apply these results to multiplicative monoids of directly indecomposable unital rings, including connected commutative rings, and to arithmetic examples arising from residue-class rings.

The $F$-invariant in cluster algebras is a combinatorial invariant that unifies the $E$-invariant from additive categorification and the $\mathfrak{d}$-invariant from monoidal categorification. In this paper, we study its refinement, the partial $F$-invariant, and establish its mutation formula under changes of the initial seed. As an application, we prove a conjecture of Reading, which asserts that the non-compatible cluster variables can be separated by sign-coherence of $g$-vectors upon varying the initial seed. We further show that, for cluster monomials, the partial $F$-invariants coincide with both the partial $E$-invariants for reachable decorated representations of quivers with potentials and the pole orders of normalized $R$-matrices (or partial $\mathfrak{d}$-invariants) for finite-dimensional reachable simple modules over quantum affine algebras. As consequences, we obtain a combinatorial formula for the pole orders for reachable simple modules in terms of $q$-characters; we verify the conjectural explicit formula for the pole orders between Kirillov–Reshetikhin modules.

Every Novikov algebra is Lie-admissible: the commutator turns it into a Lie algebra. We give a finite criterion for a multilinear element of the free Novikov algebra to belong to the Lie subalgebra generated by the free generators. By the differential realization of free Novikov algebras, the multilinear component of degree $n$ is identified with the space of homogeneous polynomials of degree $n-1$ in $n$ variables. We prove that a multilinear element is a Lie element if and only if its symbol vanishes at every integer point $(a_1,\ldots,a_n)$ with $a_i\le 1$ and $a_1+\cdots+a_n\ge 2$. Equivalently, the symbol is annihilated by two explicit linear differential operators of orders two and three. The proof combines the Witt algebra, a specialization argument for a symmetric block of variables, homogeneous interpolation and Molev's description of the multilinear component of the Lie algebra generated by the free generators as a module over the symmetric group. As applications we show that a nonzero multilinear Lie element is never a total derivative, recover the dimension of the multilinear Lie component, and illustrate the criterion by examples in degrees $4$, $5$ and $6$.