Papers

Let $k$ be a field of characteristic zero and $D_N(a_n)=a_{n+1}a_{n+2}$ the adjacent-quadratic locally nilpotent derivation on $A_N=k[a_0,\dots,a_N]$. We determine the invariant ring at $N=5$, where deleting $a_0$ gives the polynomial base kernel $B^{D_5}=K=k[p,q,J_1,J_2]$, so localization gives the polynomial-line invariant algebra $(A_5^{D_5})_{pq}=K_{pq}[W]$ with $W=F_0/(p^2q^4)$, but this does not by itself determine the global kernel. An invariant $H$ of weight $24$ and $a_0$-degree $2$ lies outside $K[F_0]$, and adjoining it gives the six-generator hypersurface $C_5$; the obstruction is explained by algebraic residue behavior defeating naive denominator descent. An invariant $F_{46}^{int}$ of weight $46$ and $a_0$-degree $4$ lies outside $C_5$ and satisfies $p^2F_{46}^{int}\in C_5$. To pass from this candidate to the full invariant ring, a second slice chart arising from $f_{13}$, together with exact $q$- and $p$-saturation, yields $A_5^{D_5}=C_5[F_{46}^{int}]$, hence finite generation, with a codimension-two complete-intersection presentation and explicit Hilbert series. At $N=6$ the transported $B$-level ring is exact, while the $A_6$ analysis is established through total degree $13$ and the full $A_6$ invariant ring remains open.

We undertake a systematic study of multiplicative monoid morphisms between prime-power modular monoids $(\mathbb{Z}/p^e\mathbb{Z},\cdot)$ and $(\mathbb{Z}/q^f\mathbb{Z},\cdot)$, for arbitrary primes $p,q$ and integers $e,f\geq 1$. Our approach proceeds in three stages. We first determine the hom-set $\mathbf{Hom}(U_{p^e},U_{q^f})$ of group morphisms between the corresponding unit groups and exhibit explicit isomorphisms describing its group structure. We then identify the subset $\mathcal{H}_{p^e,q^f,v}$ of unit group morphisms that extend to full monoid morphisms on $(\mathbb{Z}/p^e\mathbb{Z},\cdot)$, characterizing it via explicit divisibility conditions and describing its algebraic structure. Finally, we give a complete description of $\mathbf{Hom}_{\mathbf{Mon}}((\mathbb{Z}/p^e\mathbb{Z},\cdot), (\mathbb{Z}/q^f\mathbb{Z},\cdot))$ and establish an explicit monoid isomorphism between this hom-set and a naturally defined monoid built from unit groups of lower-level modular rings.

Determining whether an algebraic number field admits a power integral basis is a classical problem, but it can be difficult for fields defined by polynomial compositions and dynamical iterates. In this paper, we study the monogeneity of compositions $f(h(x))$, where $f(x)$ and $h(x)$ are monic polynomials over an arbitrary Krull valuation ring and $h(x)$ is a trinomial. We derive explicit formulas for the discriminant of the composition and use them to characterize when $f(h(x))$ generates a monogenic field. In particular, we relate the monogeneity of the composition to that of $f(x)$ and to explicit square-free conditions on the critical values of $h(x)$. We further extend the results to polynomial iteration and obtain a criterion for the monogeneity of binomial iterates, yielding new infinite families of monogenic fields. Finally, we give quantitative results and illustrate our criteria with several examples.

We show that, unlike the lower central series, the dimension series of a Lie ring need not stabilize when two consecutive terms coincide; in fact, arbitrarily long finite plateaux occur. We prove that $D_n(L)/\gamma_{n+1}(L)$ is central in $L/\gamma_{n+1}(L)$ for every $n\geq1$, in contrast to the group case. If $c\geq2$ and $L$ is metabelian and nilpotent of class at most $c$, we prove that $D_{2c-1}(L)=0$. In particular, our results imply that $D_5(L)\subseteq\gamma_4(L)$ for every Lie ring $L$.

In this article, we study the invertibility of matrices of the form $A+EDF$, where $A$ and $D$ are square matrices over a unital, not necessarily commutative, ring and $A$ is singular. Under natural hypotheses on kernels and images, we prove that $A+EDF$ is invertible if and only if $D$ is invertible, and obtain an explicit formula for the inverse of $A+EDF$ by purely algebraic methods. When the coefficient ring is stably finite, or more specifically a field, we obtain an invertibility criterion under weaker hypotheses. Our results extend and sharpen a theorem of Eriksson and Nordqvist for complex matrices.

Let $W_{\geq -1} = \mathbb{C}[t]\partial$ and $W = \mathbb{C}[t, t^{-1}]\partial$ be the Witt algebra of algebraic vector fields on $\mathbb{C}$ and $\mathbb{C}^*$ respectively. In this paper, we make significant progress toward the open conjecture that the enveloping algebras $\mathrm{U}(W_{\geq -1})$ and $\mathrm{U}(W)$ satisfy the ascending chain condition (ACC) on two-sided ideals. We show that all homomorphic images of $\mathrm{U}(W_{\geq -1})$ and $\mathrm{U}(W)$ under the family of “orbit homomorphisms” of arbitrary Gelfand-Kirillov dimension satisfy ACC on ideals. These orbit homomorphisms were the key ingredient allowing us to “lift” the Dixmier map from finite-dimensional solvable settings to infinite-dimensional settings of the Witt and Virasoro algebras in our recent work [Pham, 2025, arXiv:2504.14670]. As a result, we completely classify the prime and primitive spectra of these homomorphic images. As these images approximate $\mathrm{U}(W_{\geq -1})$ better as their GK-dimension increases, this classification sheds new light on the two-sided and prime ideal structures of $\mathrm{U}(W_{\geq -1})$. Finally, we discuss several applications of our results to the Dixmier map for $W_{\geq -1}$.

We study the Lie algebra $L_{\ge2}$ of divergence-free polynomial vector fields on $k^n$, $n\ge3$, with coefficients of degree at least two, graded by coefficient degree. Over every field its derived algebra in degree $d\ge3$ is the space of exact fields, those whose contraction with the volume form is an exact form, and it is already spanned by brackets with quadratic fields. In characteristic zero this is the whole degree-$d$ component. In characteristic $p>0$ the abelianization is nonzero above degree two exactly in the degrees $d\ge(p-1)(n-1)$ with $d\equiv1-n\pmod p$, and Cartier descent identifies it with a Frobenius twist of a rational $\mathrm{GL}_n$-module, tensored with a power of the determinant. For $p\ge5$ each exact component is obtained from the previous one by bracketing with quadratic fields.

Jurij Volčič conjectured that a noncommutative polynomial $g$ belongs to the two-sided ideal generated by $f_1,\ldots,f_l$ if and only if, for matrices of every size, the rank of the evaluation of $g$ is bounded by a constant times the maximum rank of the evaluations of $f_1,\ldots,f_l$. In this paper, we show this equivalence when the generators $f_1,\ldots,f_l$ are homogeneous. We then give an explicit counterexample in the non-homogeneous case.

We establish that, for a binary operad Var, the Hadamard product of Var and the operad Nov of Novikov algebras is a binary operad (i.e., coincides with their Manin white product Nov $\circ$ Var) if and only if the quadratic part of Var governs a 2-variety. The condition Nov $\circ$ Var $=$ Nov $\otimes$ Var is known to be equivalent to the existence of an easily verifiable characterization of those differential Var-polynomials which lie in the free derived Var-algebra. For a binary quadratic operad, these conditions are also equivalent to the Dong property of formal distributions. We distinguish the white product defined as a generated sub-operad from its quadratic approximation and give an example in which the canonical epimorphism between them is not injective.

For a third-order symmetric tensor, the Comon rank gap is the difference between its CP rank and its symmetric rank. We study direct sums of Lovitz's rational $27$-dimensional tensor, whose two ranks are $55$ and $56$. Two copies have ranks $110$ and $112$, and three copies have ranks $165$ and $168$, over both the real and the complex fields. Thus direct sums yield concise cubic tensors on spaces of dimensions $54$ and $81$ with Comon rank gaps two and three.

This article provides a complete algebraic description of $\frac{1}{2}$-derivations, local $\frac{1}{2}$-derivations, and 2-local $\frac{1}{2}$-derivations on $n$-dimensional totally graded complex filiform Lie algebras of maximum length. Based on the foundational classification framework established by Janez Bernik (2020), we systematically determine the vector spaces of $\frac{1}{2}$-derivations for the six infinite structural sequences ($m_0(n)$, $m_2(n)$, $W^+(n)$, $m_{0,1}(n)$, $m_{0,2}(n)$, $m_{0,3}(n)$) and the five exceptional one-parameter families ($g_{7,\alpha}$ through $g_{11,\alpha}$). By analyzing the pointwise local evaluation equations via parametric matrix systems, we establish the structural linearity and rigidity of local $\frac{1}{2}$-derivations. In contrast, we demonstrate that the independent parameters residing in the boundary rows of the $\frac{1}{2}$-derivation matrices provide sufficient degrees of freedom to bypass linearity constraints. Exploiting these boundary configurations, we explicitly construct pure non-linear and non-additive 2-local $\frac{1}{2}$-derivations leveraging the homogeneous function of degree one, $f(z_1, z_2) = z_1^3 / (z_1^2 + z_2^2)$, thereby defining the exact boundary where local rigidity fails.

We construct an explicit Hermitian polynomial in six selfadjoint variables, $f=-1+\omega+\omega^*+M\sum_{j=1}^{15}(2-u_j-u_j^*)$, with integer coefficients, degree $72$, and exactly $33$ monomials. Here $\omega,u_1,\ldots,u_{15}$ are specified words, $*$ reverses words, and $M$ is a specified positive integer. Its normalized trace is at least $3/4$ on selfadjoint matrix contraction tuples of every dimension, but equals $-1$ at a specified tuple of selfadjoint unitaries in a group von Neumann algebra. Thus $f$ is a counterexample to the algebraic formulation of Connes' embedding conjecture. We also construct a Hermitian quartic in $37$ selfadjoint variables, with coefficients in $\mathbb{Z}[i]=\mathbb{Z}+i\mathbb{Z}$ and $387$ monomials, attaining the same trace bounds without norm restrictions on selfadjoint matrix inputs. Degree four is minimal in this unrestricted setting. Finally, an encoding in two selfadjoint variables yields a counterexample on the contraction domain with integer coefficients and degree at most $48$; two variables are minimal.

Every scalar ordinary differential equation of order at least three with rational right-hand side that is locally linearizable by a point transformation admits a linear form with rational coefficients over the same coefficient field. We prove this by restricting the derived symmetry algebra to a coordinate line and recovering a scalar differential operator from rational symmetry-jet data. The construction uses differential elimination and linear algebra; it does not require the symmetry generators or a linearizing transformation to be solved for. A single integer parameter suffices to choose the line.

Let $A$ be a string algebra. We identify its Gorenstein-projective support $\tau$-tilting pairs with a distinguished class of maximal good collections and give a finite compatibility graph for them. Under the G-condition, we construct an explicit labelled tiled-surface model for $\mathrm{End}_A(T)$ for every nonzero basic Gorenstein-projective module $T$. Under the same condition, we show that $A$ is representation-finite if and only if $\End_A(T)$ is representation-finite for every basic Gorenstein-projective module $T$.

We introduce source-anchored algebraic amenability for Hausdorff ample groupoids, a Følner condition formulated in the Steinberg algebra and localized at compact open subsets of the unit space. The unanchored bisectional condition is exactly algebraic amenability of the Steinberg algebra, whereas the anchored version recovers ordinary amenability for one-unit groupoids and detects behavior invisible to the unanchored theory. We establish permanence under isomorphisms, open invariant reductions, disjoint unions, finite products, directed unions and AF approximations, while arbitrary subgroupoids need not inherit the property. For countable transitive discrete groupoids, infinite unit spaces are amenable through range escape, whereas finite unit spaces are amenable precisely when their isotropy groups are. This yields a characterization of hereditary algebraic amenability in terms of isotropy.

Børve, Horiatakis, and Kalck defined the magnitude of the module category of a representation-finite algebra as the sum of the entries of the inverse of the matrix of Hom dimensions between indecomposable modules. They showed it is an Euler characteristic, the alternating count of vertices, arrows, and meshes of the Auslander–Reiten quiver, and conjectured that it is at least the number of simple modules, with equality precisely for special biserial algebras. We prove this conjecture over an algebraically closed field. For a representation-directed algebra, we relate the magnitudes of the module category, the subcategory of modules without a given simple composition factor, and the ideal quotient by it. By the theory of hammocks of Ringel and Vossieck, this quotient is equivalent to a category of representations of a finite poset, whose magnitude we compute. The general case reduces to the representation-directed case through algebras obtained by grading a standard form.

For a module \(M\) over a cocommutative Hopf algebra \(\mathscr A\), we define extensive amenability by amenability of its symmetric algebra as a module over \(\operatorname{Sym}(M)\#\mathscr A\). Using the author's coalgebraic rounding and quotient theorems, we prove that nonzero extensively amenable modules are amenable, that every module over an amenable Hopf algebra is extensively amenable, and that extensive amenability is preserved and reflected by short exact sequences. For permutation modules this definition agrees, over every field, with extensive amenability of group actions. A final section gives the modifications for exterior algebras and establishes the comparison between the two definitions in several cases.

We study exact sequences of representation categories of weak Hopf algebras over an arbitrary field. Given a sequence $A\overset{k}{\to} B\overset{\pi}{\to} H$, where $A$ and $B$ are weak Hopf algebras and $H$ is a Hopf algebra, we develop verifiable algebraic conditions on $k$ and $\pi$ under which there is an exact sequence $\mathrm{Rep}(H)\to\mathrm{Rep}(B)\to\mathrm{Rep}(A)$ of tensor categories in the sense of Bruguières and Natale (2011). Along the way, we develop a generalization of the restriction of scalars functor for maps $\pi:B\to H$ between associative algebras satisfying a weakened multiplicativity constraint depending on a relatively separable subalgebra $B_r\subseteq B$, as well as a theory of kernels and cokernels for weak Hopf algebras. We, in particular, find that the cokernel of a weak Hopf algebra homomorphism $k:A\to B$ always exists, is a Hopf algebra, and the cokernel map is surjective when $A$ is connected. We conclude by studying examples of such exact sequences built from groupoids, formal ribbon extensions of quasitriangular weak Hopf algebras, and cocycled crossed products of a Hopf algebra acting weakly on a weak Hopf algebra.

We review the discrete theory of differential-geometric Poisson brackets as introduced by B. A. Dubrovin, and we present a complete proof of their characterisation in the non-degenerate case. We use this characterisation to show several novel examples of such structures, providing a complete classification for non-degenerate four-dimensional differential-geometric Poisson brackets under the additional assumption of the Lie algebra to be of quasi-Frobenius type. We also present a complete classification of quasi-Frobenius filiform N-graded Lie algebras, including two families in arbitrary dimensions.

We prove a permanental analogue of Koteljanskii's inequality. If $A$ is an inverse $M$-matrix that becomes symmetric after a positive diagonal similarity, then $\mathrm{per}(A_{S\cup T})\,\mathrm{per}(A_{S\cap T})\ge\mathrm{per}(A_S)\,\mathrm{per}(A_T)$ for all $S,T\subseteq[n]$, where $A_S$ is the principal submatrix indexed by $S$. The proof expresses permanents as moments of a complex Gaussian vector and uses Ginibre's correlation inequality. Finally, an explicit counterexample shows that symmetry cannot be dropped.

Let $B$ and $D$ be both algebras and coalgebras in a braided monoidal category $\mathcal{C}$. In the literature, there are equivalent conditions for $(B,D)$ to form a generalized smash biproduct (or cross product) denoted by $B\times D$, which were mainly given by Bespalov and Drabant in 1999 as well as by Bulacu, Caenepeel and Torrecillas in 2013. This paper is devoted to constructing another biproduct $D\times B^\ast$ when $B$ has a left dual object $B^\ast$ in $\mathcal{C}$. As results, we show that $B\times D$ is left smash over $D\times B^\ast$, and establish a specific isomorphism ${}_{B\times D}\mathfrak{YD}(\mathcal{C})^{B\times D}\cong{}_{D\times B^\ast}\mathfrak{YD}(\mathcal{C})^{D\times B^\ast}$ between the categories of the (left-right) Yetter-Drinfeld modules in $\mathcal{C}$. The constructions and results are provided in three levels: (1) $B\times D$ is an algebra and a coalgebra; (2) $B\times D$ is a bialgebra; (3) $B\times D$ is a Hopf algebra.

We consider the 653 equational laws of order up to 4 for an associative binary operation, and all of their conjunctions. We determine that there are only 456 equivalence classes of such conjunctions (associative equational theories), and find all implications between them. The conjunction operation makes this set of theories into a semi-lattice. These results are formalized in Lean. This is a semigroup analogue of the Equational Theories Project, but extended to conjunctions of equations.

The Finitistic Dimension Conjecture is one of the most important homological conjectures in the representation theory of finite-dimensional algebras. Recently, Gélinas arXiv:2004.04828 proposed the notion of "delooping level" to study the finitistic dimension Conjecture. This article reports a misconception between the delooping level and the tensor product of algebras. Namely, for finite-dimensional algebras $A$ and $B$, the difference $\operatorname{dell}(A\otimes_{\mathbf{k}}B) - \left(\operatorname{dell}(A)+ \operatorname{dell}(B)\right)$ can be arbitrarily large or negative.

Tilting mutation provides a natural way to construct derived-equivalent algebras by replacing an indecomposable summand of a tilting object. Holm, Schröer and Zimmermann~\cite{HSZ01} (see also the appendix of~\cite{AA08}) have described combinatorial transformations of the quiver and relations of a gentle algebra that yield derived equivalences via tilting complexes. In this paper, we develop a self-contained module-theoretic framework for tilting mutation via generalized $\mathrm{BB}$-tilting modules. For a gentle algebra $A=\mathbb{k} Q/\langle I\rangle$ and a vertex $k\in Q_0$, we first give a necessary and sufficient condition, expressed in terms of the arrows and relations incident with $k$, for the corresponding minimal left approximation of $P(k)$ to yield a tilting mutation. This criterion applies uniformly to vertices with or without loops. When the mutation exists, we determine the irreducible morphisms between the indecomposable summands of the mutated tilting module and use them to construct explicitly the Gabriel quiver and defining relations of its endomorphism algebra. In particular, we also provide a combinatorial mutation $(Q,I)\mapsto(Q',I')$ of gentle pairs such that $\mu_k^+(A)\cong \mathbb{k} Q'/\langle I'\rangle,$ so the resulting algebra is again gentle and derived equivalent to $A$. Finally, we formulate the dual cotilting mutation and its combinatorial description via opposite gentle pairs.

In this paper, we study various aspects of perspectivity in modules. We prove that the following classes of modules satisfy the perspective-Schröder-Bernstein property: modules with transitive perspectivity, quasi-continuous modules, Harada (and hence discrete) modules and quasi-discrete modules with the (finite) exchange property. Furthermore, we prove that for a semiregular ring, the Schröder-Bernstein property implies the perspective-Schröder-Bernstein property. We prove that for $A,B \subseteq ^{\oplus} M$, if all complements of $A$ are perspective with $B$, then all complements of $B$ are perspective with $A$. We also provide new characterizations of weakly perspective modules and modules in which perspectivity is transitive. Some applications of these results are given. We finally prove that perspectivity is transitive in a ring $R$ if and only if every special clean element of $R$ is perspective.

We study equivalences between quotient categories of module categories associated with Gabriel topologies. We establish necessary and sufficient conditions for a Morita context to induce an equivalence between quotient categories and characterize such equivalences in terms of Morita contexts after passing to rings of quotients. Motivated by Artin–Zhang's noncommutative Serre theorem, we introduce a notion of ampleness relative to a Gabriel topology and use it to characterize quotient categories of finitely generated modules under suitable noetherian hypotheses. As an application, we establish a sufficient condition under which the noncommutative Auslander theorem holds for actions of finite-dimensional Hopf algebras on AS-regular algebras without assuming that the Hopf algebras are semisimple.

Let \(\cB\) be an abelian category with enough projective and injective objects, and let $ \cA=\cB\ltimes_\eta\sF $ be an \(\eta\)-extension induced by a right exact endofunctor \(\sF\). We introduce uniform image-nilpotence for \(\eta\)-extensions and characterize it in terms of the nilpotence of the associative natural transformation \(\eta:\sF^2\to\sF\). Using the associated \(\eta\)-Loewy filtration, we prove a lifting theorem for right \(n\)-cotorsion pairs over image-nilpotent \(\eta\)-extensions and establish a corresponding completeness theorem. As applications, our results recover and unify related constructions for split nilpotent ring extensions, comma categories, formal triangular matrix rings, Morita context rings and classical trivial extensions.

We give the geometric classification of complex 3-dimensional Novikov-Poisson and Gelfand-Dorfman algebras. In particular, we prove that the variety of complex 3-dimensional Novikov-Poisson algebras is 12-dimensional; it has 4 irreducible components and has no rigid algebras; the variety of complex 3-dimensional Gelfand-Dorfman algebras is 11-dimensional; it has 10 irreducible components and has 3 rigid algebras. As an additional result, we have the geometric classification of complex 3-dimensional Novikov algebras: it is 9-dimensional, and it has $4$ irreducible components with one rigid algebra.

Let $A$ be a finite-dimensional local algebra over a splitting field. We prove that an irreducible tensor factorization $A \cong P_1 \otimes \dotsb \otimes P_n$ of the algebra $A$ is unique up to isomorphism and order of $P_1, \dotsc, P_n$.

We study Ore extensions $R[t;\sigma,\delta]$ of rings graded by an arbitrary group $\Gamma$. The grading of $R$ extends to a grading with $t$ homogeneous of degree $\gamma$ if and only if $\sigma$ and $\delta$ are compatible with conjugation and translation by $\gamma$, and this grading is then unique. For nonabelian $\Gamma$ the degree of $t$ need not centralize the support of $R$; this twist can be removed by a change of variable when $R_\gamma$ contains a unit, but not in general. The graded forms of the classical necessary conditions for simplicity are not sufficient, as the quantized Weyl algebra shows. If $R$ is graded-simple and $\sigma$ is an automorphism, then $R[t;\sigma,\delta]$ is graded-simple if and only if it contains no homogeneous normal element that is monic of positive $t$-degree. For Ore extensions of group algebras, an affine action of the group on the line describes all graded ideals; in characteristic zero, graded simplicity is equivalent to the nonvanishing of a class in $H^1(G,k_\chi)$, and simplicity forces the group to be metabelian.