Papers

We prove that the bracket width of the Lie algebra of vector fields on any smooth affine algebraic variety of dimension $n$ is at most $(n+1)^2$. We give improved bounds for some families of $\mathbb{C}^*$-varieties, in particular for $\mathrm{SL}_n(\mathbb{C})$ and for the Koras–Russell cubic threefold.

We show that the operadic structure on the cohomology of the poset of noncrossing 2-partitions is isomorphic to the Brace operad.

Jungin Lee in 2018 proved a necessary and sufficient condition that an integral quadratic form $\sum_{i=1}^{m} a_iX_i^2$ is universal over $M_2(\mathbb{Z})$. For a positive integer $n \geq 2$, Lee defined $f(n)$ to be the smallest positive integer $m$ such that for every pairwise coprime $a_1, a_2, \ldots a_m \in \mathbb{Z}$, $\sum_{i=1}^{m}a_iX_i^2$ is universal over $M_n(\mathbb{Z})$. He gave bounds on $f(n)$ too. Koo and Lee further improved the bounds in 2025. This paper is organized as follows. Let $\mathbb{Z}_p$ be the ring of $p$-adic integers. In the first section we give necessary and sufficient condition for a quadratic form $\sum_{i=1}^{m}a_iX_i^2$ to be universal over $M_2(\mathbb{Z}_p)$ for $p=2$ and for an odd prime $p$. Consequently we express matrices of over $\mathbb{Z}_p$ as sum of squares. For a positive integer $n \geq 3$, we define $f(n)$ to be the smallest positive integer $m$ such that for every $p$-adic integers $a_1,a_2, \ldots a_m$ with at least three of them units, the diagonal quadratic form $\sum_{i=1}^{m}a_iX_i^2$ is universal over $M_n(\mathbb{Z}_p)$. In the next section we find bounds on $f(n)$ for $n \geq 3$.

We prove that for any countable ring $R$ (not necessarily commutative), if the associated left $R$-module ${}_R R$ has a strictly descending sequence of pp-definable subgroups, then the theory $Th(R^{(\omega)})$ of the infinite dimensional direct sum is Borel complete. From this, we conclude that if $R$ is countable and not left perfect, then the theory of $R$-modules is Borel complete, and we give a full characterization of which countable simple rings have Borel complete theories. One special case is that the complete theory $Th({\mathbb Z}^{(\omega)})$ is Borel complete, which strengthens the existing proofs of the Borel completeness of TFAB, the theory of torsion free abelian groups. The proof also introduces, relative to the chosen pp-chain, a proper two-sided ideal $L^R$, and a notion of f.g. hulls which, for countable rings and countable parameter sets in theories satisfying $T=T^{\aleph_0}$ exist and are unique up to isomorphism. These constructions may be of independent interest in the model theory of modules.

The class of left $h$-adequate semigroups sits between the classes of left ample and left Ehresmann semigroups. Whilst free objects in related classes have descriptions in terms of directed trees, free left $h$-adequate semigroups have yet only been described in terms of Rees quotients of free products. Here, we reintroduce the class of (left) $h$-Ehresmann semigroups, show their free objects coincide with free (left) $h$-adequate semigroups, and give a description of these free objects in terms of directed trees. We use our new description in investigating various properties of these structures, including finitary conditions of recent interest.

There is a diassociative algebra structure on the tensor product of an associative algebra and a perm algebra. In this paper, we elevate this conclusion to the level of bialgebra.We prove that the tensor product of an antisymmetric infinitesimal bialgebra and a quadratic perm algebra has a diassociative bialgebra structure, and this diassociative bialgebra structure is coboundary (resp. quasi-triangular, triangular, factorizable) if the original antisymmetric infinitesimal bialgebra is coboundary (resp. quasi-triangular,triangular, factorizable). As an application, we provide the close relationship between Lie bialgebras, Leibniz bialgebras, diassociative bialgebras and antisymmetric infinitesimal bialgebras, and the close relationship between symplectic Lie algebras, symplectic Leibniz algebras, symplectic associative algebras and symplectic diassociative algebras.

We observe that standard derived equivalences give a counterexample to a conjecture of Kalck on global dimension for weighted projective lines. For the root stack $X=\mathbb{P}^1\langle \infty,0,1;2,3,3\rangle$ we exhibit a $13$-dimensional radical-square-zero algebra $A$ such that $$ D^b(\mathrm{coh}X)\simeq D^b(\mathrm{mod}A), \qquad \mathrm{gldim}A=4>3. $$

The paper "Orthogonal completion of algebraic systems II" continues the line of "Orthogonal completion of algebraic systems" from the perspective of almost classical localizations of algebras and the conclusions of K.~I.~Beidar's doctoral dissertation. This allows one to apply the orthogonal completion method to describe non-degenerate orthogonally complete algebras bases on known classifications of strongly prime algebras.

We give an equivalent definition of a non-pointed torsion theory, also called a pretorsion theory in the literature. This uses a new, more general, notion of a near-torsion theory, and does not use non-pointed exactness. Furthermore, a general method of associating the `largest' (non-pointed) torsion theory to any given near-torsion theory is described. We also describe a near-torsion theory formed by the class of cokernel diagrams and the class of kernel diagrams in a pointed category with kernels and cokernels, and its associated torsion theory.

We employ the machinery of linear relations to the study of optimization problems in linear algebra. We first show that the relational version of the pseudo-inverse can be realized through a generalization of the least-squares problem. This allows one to prove that the pseudo-inverse realizes the solution of certain relational optimization problems. Our main result is showing that a certain truncation of this pseudo-inverse defines a solution to a relational version of the classical low-rank approximation problem which recovers both the Eckart-Young Theorem and several optimization problems involving pairs of matrices and vector spaces.

Let $A$ be a finite-dimensional representation-finite algebra over an algebraically closed field $k$, and let $M$ be a finite-dimensional $A$-module such that $\operatorname{End}_A(M)$ has finite global dimension. We prove that $\operatorname{End}_A(M)$ has Cartan determinant one if $A$ has finite global dimension or $\operatorname{char}k\neq2$; in particular, the Cartan determinant conjecture holds for representation-finite algebras over an algebraically closed field.

In this paper, we construct deterministic matrices from subspaces of orthogonal spaces over finite fields of odd characteristic and investigate their applicability to compressed sensing. The construction is based on incidence relations among three types of subspaces, yielding families of matrices with explicitly computable dimensions and coherence. Using coherence-based estimates, we establish sufficient conditions under which these matrices satisfy the Restricted Isometry Property for prescribed sparsity levels. We also provide numerical comparisons with DeVore's deterministic construction to illustrate the trade-off between the number of measurements, coherence, and sparse recovery guarantees.

In this paper, we provide a complete classification of real three-dimensional simple Bol algebras $(B, [.,.], \langle .,.,. \rangle )$. The main part of the classification concerns the case $[B,B]=B$. In addition, we identify two isomorphism classes with a nonzero binary product, both satisfying $\dim[B,B]=2$, as well as four simple classes of Lie triple systems in which the binary product vanishes.

We classify, up to PI-equivalence, associative algebras over a field of characteristic zero whose codimension sequences have quadratic growth. More precisely, making use of fundamental algebras, we prove that every such algebra is PI-equivalent to a finite direct sum of algebras generating minimal varieties of at most quadratic codimension growth, together with a possible nilpotent summand.

Stone (Ph.D.\ thesis, Department of Operations Research, Stanford University, 1981) proved that every matrix in $U \cap Q_0$ is a $P_0$-matrix and conjectured that the same conclusion holds for the larger class $E_0^f \cap Q_0$ of fully semimonotone $Q_0$-matrices. Murthy and Parthasarathy [SIAM J.\ Matrix Anal.\ Appl.\ 16 (1995), 1268–1286] verified the conjecture for matrices of order up to $4 \times 4$, for $5 \times 5$ and $6 \times 6$ matrices under additional hypotheses, and for several special subclasses of arbitrary order, but the conjecture remains open in general. In this paper we prove that every $E_0^f$-matrix with positive determinant is a $P_0$-matrix, for matrices of arbitrary order $n$; our proof proceeds by induction on $n$, via an algebraic analysis of principal minors under principal pivotal transforms. We further exhibit a matrix $A \in E_0^f$ with $\det A > 0$ that fails to belong to $Q_0$, showing that the hypothesis $\det A > 0$ used in our theorem cannot, by itself, be deduced from membership in $Q_0$, and hence does not on its own yield a proof of Stone's conjecture. Stone's conjecture itself remains open.

The uniform car-parking process is the one-dimensional random sequential adsorption of unit cars on a segment of finite length $s$: cars arrive at uniformly random positions and park wherever they fit, until no gap admits another. This paper develops the exact finite-$s$ theory. The joint density of the parked positions is resolved into jamming cells, on each of which it is a rational function, and evaluated by a subset recursion in $O(2^n n)$ operations; the marginal and gap order statistics are obtained as hyperlogarithms whose weight is fixed by the number of coordinates integrated out; and the absorption count and the aggregate quantities are treated through the integral equation descending from Rényi.

We study the Marshall quotient \[ M(n)=M(\mathbb Z/n\mathbb Z) \] obtained from the ring of integers modulo $n$ by quotienting by the square classes of non-zero-divisors. Using elementary arithmetic of square classes modulo prime powers and the Chinese Remainder Theorem, we give an explicit description of these quotients and classify several of their structural properties. We determine when the quotient relation is arithmetically elementary, when $M(n)$ is hyperbolic, when it can be real reduced, and when it can be formally real. We also analyze the subset of invertible classes together with zero, proving exactly when it is a submultiring, when it is a hyperfield, and when it is hyperbolic. The results provide a finite family of test examples for questions connecting multirings, special hyperfields, real semigroups, and abstract quadratic-form theory.

In this paper we present a generalization of Jordan pairs, called Jordan pair disystem, which appear naturally as the wings of a Leibniz algebra with a finite $\mathbb{Z}$-grading; conversely, we give the Tits-Kantor-Koecher construction for Jordan pair disystems, obtaining a Leibniz algebra with a short $\mathbb{Z}$-grading. This construction extends the construction given by Gubarev and Kolesnikov for Jordan dialgebras (arXiv:0907.1740). We introduce homotopes for Jordan pair disystems at elements and we show that we obtain again a Jordan dialgebra. Moreover, we introduce abelian inner ideals and their kernels for Leibniz algebras, and prove that the subquotient of a Leibniz algebra with respect to an abelian inner ideal is a Jordan pair disystem. This notion of subquotient extends the construction of Jordan dialgebras at $Q$-Jordan elements given by Felipe and Velásquez.

In this paper, we introduce the notions of extended $\mathcal{O}$-operators on perm algebras, as generalization of $\mathcal{O}$-operators, and the extended perm Yang-Baxter equation. We demonstrate that an $\mathcal{O}$-operator of weight $\lambda$ on a perm algebra gives rise to a post-perm algebra, and an extended $\mathcal{O}$-operator yields a new perm algebra structure. We also give an equivalent characterization of extended $\mathcal{O}$-operators in terms of ordinary $\mathcal{O}$-operators via the $\pi_\pm$ decomposition. The notion of the generalized perm Yang-Baxter equations is introduced, and their relationship with extended $\mathcal{O}$-operators is established. The tensor form of extended $\mathcal{O}$-operators leads to the notion of the extended perm Yang-Baxter equation (extended perm-YBE), which generalizes the perm Yang-Baxter equation. We establish that a solution of the extended perm-YBE with $(R, \ad)$-invariant skew-symmetric part is characterized by an extended $\mathcal{O}$-operator. Furthermore, relationships among extended $\mathcal{O}$-operators, the perm Yang-Baxter equation and the extended perm-YBE are investigated through the framework of quadratic perm algebras and semi-direct product perm algebras.

Let $H$ be a weak Hopf algebra with bijective antipode. This paper is devoted to the AS-Gorenstein property of restricted faithfully flat weak $H$-Galois extensions $A/B$. We first of all study the cohomologies of weak $H$-Galois extensions and then establish the spectral sequence connecting the Hochschild cohomologies of $A$ and of $B$. Finally we prove that when $A$ is a noetherian affine PI algebra and $B$ is AS-Gorenstein, $A$ shares the AS-Gorenstein property.

The construction problem of Leibniz bialgebras from Lie bialgebras and perm bialgebras is considered in this paper. We show that there is a Leibniz algebra structure on the tensor product of a Lie algebra and a perm algebra, and elevate this conclusion to the level of bialgebra. We prove that the tensor product of a quadratic Lie algebra and a perm bialgebra has a Leibniz bialgebra structure, and this Leibniz bialgebra structure is coboundary (resp. quasi-triangular, triangular, factorizable) if the original perm bialgebra is coboundary (resp. quasi-triangular, triangular, factorizable). Moreover, we constructed an infinite-dimensional Leibniz bialgebra using the tensor product of a finite-dimensional Lie bialgebra and a quadratic $\bz$-graded perm algebra. Quasi-triangular and triangular infinite-dimensional Leibniz bialgebras are considered.

Let $0\leq k < n$, and let $\mathcal V\subseteq M_n(\mathbb C)$ be a complex linear subspace satisfying $\operatorname{rank}[S,T]\leq k$ for all $S,T\in\mathcal V$. Omladič, Radjavi, and Šivic proved the sharp bound $\dim\mathcal V\leq nk+\left\lfloor (n-k)^2/4\right\rfloor+1$ and conjectured a classification of the equality cases. We prove their conjecture. If equality holds, then, after a similarity and possibly transposition, $\mathcal V$ consists of all block upper-triangular matrices with arbitrary upper-left and upper-right blocks and with lower-right block in a maximal-dimensional commuting subspace of $M_{n-k}(\mathbb C)$. For $n-k\geq4$, these commuting subspaces are the classical equality cases in Schur's theorem; in dimensions $2$ and $3$, the additional equality cases also occur. At the equality dimension, the rank condition defines a projective algebraic subset of a Grassmannian. For $2\leq k\leq n-2$, we determine all of its irreducible components. If $n-k\geq4$, there are exactly two components when $n-k$ is even and exactly four when $n-k$ is odd; if $n-k\in\{2,3\}$, there are exactly two. When $n-k\geq4$, the Zariski tangent space at each such space equals the tangent space to its conjugacy orbit. When $n-k\in\{2,3\}$, the two components are obtained by varying the invariant $k$-dimensional subspace and the maximal-dimensional commuting subspace on the quotient, together with their transposes.

In this paper, we establish a necessary and sufficient criterion for a finite-dimensional semisimple algebra over an algebraically closed field of characteristic $0$ to admit a regular quantum commutative decomposition. We apply this characterization to group algebras and show that a finite group algebra admits such a decomposition if and only if the underlying group is a peak group, that is, a finite group whose irreducible character degrees have a greatest element with respect to the divisibility ordering. We investigate the structure of peak groups and establish several criteria for their solvability in terms of the prime divisors of their largest irreducible character degree. In particular, we show that several important classes of groups are peak groups, while supersolvability alone does not imply the peak property. Finally, we construct a non-solvable peak group within the class of Frobenius groups.

It follows by Puuska that, over a commutative Noetherian ring $R$, a morphism $i : M \rightarrow I$ of $R$-modules is an injective envelope if and only if its Matlis dual $Hom_R(i, E)$ is a flat cover for some injective cogenerator $E$, and equivalently for every injective $R$-module $E$. In this paper, we will generalize this result to non-commutative Noetherian algebras.

We investigate identities satisfied by monoids of triangular matrices over fields and over additively idempotent semirings with 0 and 1, and exhibit several new instances in which these identities admit no finite axiomatization.

We show that for any real, connected, semi-simple matrix group, its Lie algebra of algebraic vector fields can be generated by three complete vector fields.

We parametrize the isometry and equivalence classes of skew constacyclic codes over a finite field by classifying the corresponding classes of their ambient rings, and present algorithms for the parametrizations. We achieve a tight classification by taking all possible Hamming-weight preserving isomorphisms between their ambient Petit rings into account. We also count the number of equivalence classes of these rings. We present many examples where isometry is a strictly stronger relation than equivalence, that is, codes that are isometric but not equivalent.

Cluster varieties contain the union of cluster tori and the points not in this union are called $\textit{deep points}$, and the locus of these points is called the $\textit{deep locus}$. In arXiv:2402.16970, a description of this locus is conjectured for locally acyclic cluster algebras, in particular, stating that this should be the stabilizer locus of the cluster automorphism group. We resolve this conjecture for the case of cluster algebras arising from surfaces, introduced in arXiv:math/0608367, and the remaining finite type cases as categorized in arXiv:math/0208229. In particular, we show locally acyclic surface type cluster varieties have no deep points not contained in the stabilizer locus, and finite type cluster algebras also have no deep points not contained in the stabilizer locus, validating the mysterious points conjecture in these cases.

In a wide variety of fields, every non-scalar matrix $M$ is similar to a matrix all of whose entries are non-zero. We give extensions of this result, including some in the division ring context. We then apply these results to the question of which pairs of operators generate the algebra of all operators on finite- and infinite-dimensional spaces. In the case of complex, separable infinite-dimensional Hilbert space, this is intimately related to the unsolved Invariant Subspace Problem. Finally, linear independence of subsets of arbitrary TVS's and generators of arbitrary TA's are shown to be stable in the sense of the theorems presented.

The general linear groupoid $\mathbf{G}(L(E))$ of a Leavitt path algebra $L(E)$ is isomorphic to the groupoid whose objects are all $L(E)$-modules of the form $\bigoplus_{i=1}^n v_iL(E)$ where each $v_i$ is a vertex, and whose morphisms are all isomorphisms between these modules. We find a generating set for $\mathbf{G}(L(E))$ and consequently obtain generating sets for all general linear groups $\text{GL}_n(L(E))$ over $L(E)$ (including the group $\text{GL}_1(L(E))$ of invertible elements of $L(E)$). We prove similar results for Leavitt path algebras of hypergraphs (which generalise the Leavitt path algebras of separated graphs and vertex-weighted graphs).