Papers

Given a finite dimensional Lie algebra $L$ on a finite field $\mathbb{F}_{p^n}$ of prime power order $p^n$ (with $n$ positive integer and $p$ prime), we consider the number of modular pairs $(A,B)$ in the lattice of all subalgebras $\mathcal{L}(L)$ and introduce the notion of “subalgebra commutativity degree” of $L$. This represents the probability to find that two randomly chosen subalgebras $A$ and $B$ of $L$ are permutable. We investigate the subalgebra commutativity degree of $L$ in connection with recent techniques of algebraic combinatorics and number theory, providing upper and lower bounds which may influence the structure of $L$. A specific study for the subalgebra commutativity degree of Heisenberg algebras is executed.

Fu, Guil Asensio, Herzog and Torrecillas asked whether a complete ideal cotorsion pair of object ideals in an exact category always induces a complete cotorsion pair of objects. We show that it need not, even in a Hom-finite, weakly idempotent complete Frobenius exact category. Our example is built from bounded complexes of finite-dimensional vector spaces with even total cohomology dimension. The parity obstruction manifests itself as a non-split idempotent in the stable category. For cotorsion pairs of $t$-structure type, we give an object-wise splitting criterion for special approximations. In this setting, a criterion of Saorín and Šťovíček yields an affirmative answer whenever the stable category is idempotent complete.

In this paper, we consider three growth functions of algebras: the Gelfand-Kirillov dimension, superdimension, and entropy – as well as a variation of the latter successfully used in the study of Leavitt path algebras. We prove results that make precise the heuristic fact that the Gelfand-Kirillov dimension is suitable for the study of algebras with polynomial growth, the superdimension for algebras with subexponential growth, and the entropy for algebras with exponential growth. We introduce a definition of entropy for graded modules motivated by the entropy for graded algebras. As an application of our study, we give a new criterion for Leavitt path algebras to be PI.

Fuzzy hypervector spaces provide a structural framework for modeling graded uncertainty in algebraic systems with multivalued operations. In this paper, we introduce and investigate admissible mappings as a constructive mechanism for generating fuzzy subhyperspaces. These mappings represent fuzzy substructures through families of fuzzy points and allow a systematic characterization of generators. We prove that maximal admissible mappings generate fuzzy subhyperspaces in a manner analogous to bases in classical linear theory, and we present examples over real and finite hypervector spaces to clarify the role of maximality. The proposed approach contributes to the structural foundations of fuzzy modeling in hyperalgebraic settings and may support further developments in soft computing environments involving graded and non-deterministic structures.

The derived delooping level is a recently introduced homological invariant that provides an upper bound for the finitistic dimension of the opposite algebra. In this paper, we employ two reduction techniques-cleft extensions and recollements-to study the finiteness of the derived delooping level of finite-dimensional algebras over a field. By applying the theory of cleft extensions to bound quiver algebras, we establish arrow-removal operations that preserve the finiteness of the derived delooping level. In parallel, using recollement techniques, we develop vertex-removal operations with the same finiteness-preserving property. We conclude with several examples illustrating the applicability and effectiveness of these reduction methods.

In this paper, we further investigate the role of the graded Grothendieck group $K_0^{\gr}$ and its positive cone (the talented monoid) as an effective tool for distinguishing structural types of algebras associated to higher-rank $k$-graphs. We study the prime spectrum (the space of all prime $\Gamma$-order ideals equipped with a Zariski-like topology) of a general commutative $\Gamma$-monoid and establish that this space is spectral in the sense of Hochster, provided the monoid has the refinement property and every $\Gamma$-order ideal is finitely generated. As a result we are able to show that the prime spectrum of the talented monoid of a row-finite $k$-graph without sources and with a finite set of vertices is spectral. For any row-finite $k$-graph $\Lambda$ without sources, one of our main results says that the space of all graded prime ideals of the Kumjian–Pask algebra $\KP(\Lambda)$ is homeomorphic to both the space of all prime $\mathbb{Z}^k$-order ideals and the space of all prime $\mathbb{Z}^k$-filters of the talented monoid $T_\Lambda$. Another main result of this paper provides a complete topological description of regular $\Gamma$-order ideals of a refinement $\Gamma$-monoid: a $\Gamma$-order ideal $J$ is regular if and only if the corresponding closed (resp., open) set $V(J)$ (resp., $D(J)$) is regular closed (resp., regular open) in the prime spectrum. As an application of these results, we establish a lattice isomorphism between the lattice of all regular $\mathbb{Z}^k$-order ideals of the talented monoid and the lattice of all regular graded ideals of the Kumjian–Pask algebra via a spectrum-theoretic approach.

We study the minion homomorphism poset of multisorted Boolean clones. Our main result completely determines the subposet induced by products of Boolean clones: every inequality between prod- ucts is induced by inequalities between their Boolean factors. Furthermore, we determine the minion cores of all products of Boolean clones, thereby identifying canonical representatives of their minion homomorphic equivalence classes.

We prove that the Auslander–Reiten conjecture, the generalised Nakayama conjecture, the Nakayama conjecture, and each of the two Tachikawa conjectures are globally equivalent for Artin algebras over any fixed commutative artinian ring. For finite dimensional algebras over a fixed field, we also prove that the derived invariance of the Nakayama conjecture is equivalent to the truth of the Nakayama conjecture for all such algebras, answering a question raised by Chen and Xi.

By addressing the problem of describing the differences between the saturation monoids of the Prandtl and Leray-$\alpha$ boundary layers, and algebraically distinguishing the action of the Helmholtz operator, we can show that, when the modification of the vector field is halted at a minimum scale determined by the differential filter, the paradox of constructing a solution that reaches infinite velocity in a finite time is avoided. Furthermore, it allows us to obtain the coefficients of the Taylor expansion near the wall for boundary layers where the Helmholtz operator has been applied.

We study the behaviour of regular sublattices of infinitely distributive lattices under completion. Using order convergence, we associate with an infinitely distributive lattice \(L\) the complete lattice \(\mathfrak I_L\) of \(O\)-closed order ideals and describe the corresponding closure operator \(A\mapsto A^{\sigma_L}\). We show that every lattice homomorphism induces a canonical map between the corresponding lattices of \(O\)-closed ideals, and compare \(\mathfrak I_L\) with the Dedekind–MacNeille completion \(\DM(L)\). When \(\DM(L)\) remains infinitely distributive, this comparison yields extension results for lattice homomorphisms and a join-regular realization of \(\DM(Y)\) inside an ambient complete lattice. We also determine a sharp finite-dimensional obstruction. We construct an infinitely distributive regular sublattice \(L\subseteq\mathbb R^3\) whose Dedekind–MacNeille completion is not even modular, and therefore cannot be realized as a sublattice of \(\mathbb R^3\). In contrast, we prove that the Dedekind–MacNeille completion of every bounded sublattice of a product of two chains is distributive. Thus dimension three is the least dimension in which a bounded sublattice of a finite product of real chains can have a non-distributive Dedekind–MacNeille completion.

We develop a probabilistic approach to the structure of finite skew left braces, motivated both by classical commuting probability in finite group theory and by the role of skew left braces in the study of set-theoretic solutions of the Yang–Baxter equation. We introduce relative commuting probability for skew left braces and establish analogues of several classical structural results, including relative BFC-type theorems. For natural classes of finite skew left braces, we show that a positive lower bound for the commuting probability forces the existence of a large section which is trivial up to bounded subgroups. We also study the probability that two elements generate a trivial sub-skew brace and a Sylow-local version of commuting probability, obtaining further structural consequences. Finally, we introduce a commuting probability for finite non-degenerate set-theoretic solutions of the Yang–Baxter equation. Under natural hypotheses on the associated structure skew brace $G(X,r)$, a positive lower bound for this probability yields a bound on $|G(X,r):\Soc(G(X,r))|$ depending only on the probability and on $|X|$.

Semigroups in which right divisibility is a partial order, that is, $\mathscr R$-trivial semigroups, have been studied by people in algebraic combinatorics in connection with Markov chains (especially left regular bands). Margolis and the author showed that the quiver of the algebra of an $\mathscr R$-trivial monoid is field independent. We prove here that the integral semigroup ring of an $\mathscr R$-trivial semigroup has a bound quiver presentation provided that it is unital. This then gives a uniform quiver presentation over any field via extension of scalars.

We describe the ring of Laurent symmetric polynomials in terms of generators and relations, giving an analogue of the Fundamental Theorem of Symmetric Polynomials. We extend the Hall inner product, and give an algebraic proof that the Laurent Schur polynomials form an orthonormal basis of this ring. In the case of infinitely many variables, we realize the ring of Laurent symmetric functions as a direct limit of inverse limits, and relate them to characters of rational and algebraic representations of general linear groups.

We study the Cuntz semigroup of arbitrary rings with stable rank one. We prove that Cuntz subequivalence between countably generated projective right modules $P$, $Q$ is equivalent to $P$ being isomorphic to a pure submodule of $Q$, whilst Cuntz equivalence amounts to isomorphism.~We also prove order-cancellation of finitely generated projective modules (and, more generally, pairs of modules admitting certain complements) from direct sums with countably generated projective modules.

Let $(\mathscr{C}, \mathbb{E}, \mathfrak{s})$ be a weakly idempotent complete extriangulated category, and let $\xi_1\subseteq\xi_2$ be proper classes of $\mathbb{E}$-triangles. We prove an analogue of Cole's mixing theorem in this setting: two compatible admissible model structures, relative to $\xi_1$ and $\xi_2$ respectively, give rise to a mixed admissible model structure $\mathcal{M}_m$ relative to $\xi_1$, which has the trivial objects of the second model structure and the fibrant (or cofibrant) objects of the first. We also describe the cofibrant objects of $\mathcal{M}_m$ explicitly in terms of the two original model structures. For exact categories this recovers the mixing theorem for exact model structures, and for triangulated categories it applies to proper classes of triangles; we give a non-degenerate example on the homotopy category of a quasi-Frobenius ring.

The problem of determining whether a category is balanced (i.e., every morphism that is both an epimorphism and a monomorphism is an isomorphism) has been widely investigated, especially in algebra. For complete categories with an extremal projective generator, we characterise the property of being balanced, and the stronger property that every monomorphism is regular, in terms of implicit operations. These characterisations apply, for example, to any quasiequational class of algebras, possibly with a proper class of function symbols and unbounded arities, that admits free algebras.

Let $\mathbb O$ be the real octonion division algebra and $\mathfrak g_2=\operatorname{Der}(\mathbb O)$. A quaternionic reflection $T\in G_2=\operatorname{Aut}(\mathbb O)$ fixes a quaternionic subalgebra pointwise and negates its orthogonal complement. For the associated compact symmetric pair $\mathfrak g_2=\mathfrak k_T\oplus\mathfrak m_T$, we give an explicit factored polynomial criterion for generation by an independently chosen even element and odd element. Equivalently, for the reflected pair $(Z,TZT)$ with $Z\in\mathfrak g_2$, the criterion is $\mathcal U_{38}=\mathcal R_{30}\mathcal H_8>0$. The degree-30 Gram determinant detects reducibility on $\operatorname{Im}\mathbb O$, while the degree-8 factor detects generated algebras of dimension at most three. A principal $\mathfrak{su}(2)$ is the remaining irreducible proper possibility. Such a reflected principal closure is realizable for a nonzero $Z$ if and only if its positive rotation frequencies have ratio $1:2:3$; for each such $Z$, all realizing reflections are parametrized by a two-torus and an open interval. A different restriction arises when $Z$ must lie in the stabilizer algebra $\mathfrak k_A\simeq\mathfrak{so}(4)$ of a fixed quaternionic subalgebra $H_A$. For every reflection moving $H_A$, a fixed list of six elements of $\mathfrak k_A$ contains a generating choice; reflections preserving $H_A$ admit none. For one explicit relative configuration $(\mathfrak k_A,S)$, we describe the entire nongenerating set by four geometric families and by six irredundant systems of scalar equations. We determine its real dimension and all generated algebras of dimension at most three; a homogeneous degree-34 polynomial packages the six tests. The degree is not asserted to be canonical or minimal. Two explicit local coordinate maps use short words in a finite-time pulse and one reflection, whose generated subgroup is dense in $G_2$.

At every prime $p$, we disprove the algebraic generating hypothesis for compact objects in Hovey's stable category of $BP_*BP$-comodules and its local analog at every positive height, answering questions of Barthel and Heard. In both settings, for every $r,n\geq1$, we construct a ghost whose first $n$ composition powers are all nonzero and have exact additive order $p^r$. The same construction applies to the stable category of $E_*E$-comodules for even $p$-local Landweber exact theories $E$ that are not rational. Through the motivic deformation of Gheorghe–Wang–Xu, these examples give $C\tau$-linear counterexamples to the cellular $\mathbb{C}$-motivic generating hypothesis, with the same order and composition properties. We also construct nonzero non-$C\tau$-linear ghosts between $C\tau$-modules and a nonzero ghost on a motivic spectrum with noncontractible Betti realization.

For the Iwahori-Hecke algebras $\mathcal H_q(\mathfrak S_n)$ of the symmetric group $\mathfrak S_n$ at a primitive $e$-th root of unity, James and Mathas proved a theorem which relates $v$-decomposition numbers $d_{\lambda\mu}^{e}(v)$ for different values of $e$, by adding “empty runners” to the abacus display for the labelling partitions $\lambda,\mu$. Fayers proved a similar theorem, which involves adding “full” runners to these abacus displays. In this paper we use Uglov's map from the set of $r$-abaci for $r$-partitions to the set of $e$-abaci for partitions to extend these theorems to the cyclotomic Hecke algebras of type $G(r,1,n)$.

We give a short construction refuting the Köthe Conjecture. The construction is based on the point module approach to the problem, introduced by the first-named and second-named authors together with Easton and Ford.

We study the real three-dimensional unital algebra with basis $\{1,i,j\}$ and products $i^2=j^2=-1,\quad ij=0,\quad ji=i+j$. Its quadratic and power-associative elements form the union of exactly three complex subalgebras; their nonzero elements are precisely those with a two-sided inverse, which is unique. Every nonzero subalgebra contains the unit, and every element outside these three planes generates the whole algebra. We classify one-sided inverses, including the singular cases, and show that two-sided invertibility does not imply cancellation. The multiplication algebra is $M_3(\mathbb{R})$, so the algebra is absolutely central simple. Its derivations vanish, its automorphism group is trivial, and its commutator Lie algebra is $\mathbb{R}\oplus\mathfrak{aff}(1)$. Its unique antiautomorphism is involutive, and its complexification admits no complex-linear structurable involution. Finally, replacing $ji=i+j$ by $ji=t(i+j)$ gives real algebras $\mathcal{A}_t$ with $\mathcal{A}_s\cong\mathcal{A}_t$ exactly when $s^2=t^2$. We describe the exceptional parameters and thereby exhibit nontrivial deformations of the original algebra.

We show that every countable group $G$ that has Ozawa's property $\rm PHP$ and contains a non-abelian free subgroup has the following property: the group ring $KG$ is primitive for any field $K$. As a consequence, we deduce that all non-trivial countable linear groups with trivial amenable radical have this property. Along the way, we prove that the class of groups with $\rm PHP$ enjoys some permanence properties that are of independent interest.

We present a generalization of the Fast Fourier and fast Walsh transform algorithms to tensor products of $n$ arbitrary $q\times q$ matrices over a commutative ring, reducing the cost of applying such a tensor product from $q^{2n}$ to $nq^{n}$ ring operations. We then give an explicit construction of square orthogonal matrices over a commutative unitary ring, starting from a prescribed first row, and specialize this construction to ${\mathbb Z}/256{\mathbb Z}$. Combining these two ingredients, we propose a symmetric cryptosystem in which the secret key determines~$n$ orthogonal matrices over ${\mathbb Z}/256{\mathbb Z}$ whose tensor product encrypts a block of $q^n$ bytes; encryption and decryption both exploit the fast tensor product algorithm. We give three examples of implementations.

We calculate the quotient of the mod-two cohomology of a product of symmetric and alternating groups by the ideal of negligible elements, namely those that restrict trivially to any order-two subgroup. This quotient arises in the study of cohomological invariants of fields, as by work of Serre the ideal of negligible elements coincides with the ideal of elements that pull back trivially in Galois cohomology. Our calculations answer questions recently posed to us by J-P. Serre, whose recent results we build on. It is a pleasure to share these results on the occasion of his 100th birthday.

We prove that every Igusa–Todorov Artin algebra satisfies the Auslander–Reiten conjecture. More precisely, let $V$ be an $n$-Igusa–Todorov witness, and let $t$ be the number of isomorphism classes of nonprojective indecomposable summands of $V$. If a finitely generated module $M$ satisfies $\Ext_A^i(M,A)=0$ for every $i>0$ and $\Ext_A^q(M,M)=0$ for $1\leq q\leq 2t+1$, then $M$ is projective. As applications, algebras of representation dimension at most three and algebras satisfying $J^{2m+1}=0$ for which $A/J^m$ has finite representation type satisfy the Auslander–Reiten conjecture.

Let $Q$ be a finite acyclic valued quiver. We establish an alternative cluster multiplication formula in the quantum cluster algebra associated with $Q$, which exhibits certain symmetries similar to 2-Calabi–Yau properties of cluster categories. Motivated by this formula, we introduce a Hall algebra $\mathcal {D}\mathcal {H}_\Lambda^{{cl}}(\widetilde{\mathcal{A}})$ via a certain quotient of a derived Hall subalgebra associated with $Q$, serving as an algebra counterpart to cluster categories of hereditary algebras. As our main result, we prove that the quantum cluster algebra associated with $Q$ is isomorphic to the exceptional subalgebra of $\mathcal{D}\mathcal{H}_\Lambda^{cl}(\widetilde{\mathcal{A}})$, namely, the subalgebra generated by exceptional objects. Furthermore, when $Q$ is of Dynkin type, the quantum cluster algebra is naturally isomorphic to $\mathcal{D}\mathcal{H}_\Lambda^{{cl}}(\widetilde{\mathcal{A}})$ itself. This establishes a Hall algebra realization for all acyclic quantum cluster algebras. As an application, we introduce the Hall subalgebra of $\mathcal{D}\mathcal{H}_\Lambda^{cl}(\widetilde{\mathcal{A}})$ generated by projectives, yielding a Hall algebra analogue of the lower bound quantum cluster algebras. Then we show that the standard monomials of quantum projective cluster variables form a basis of the quantum cluster algebras via Hall algebra approach. Moreover, we show that the Auslander–Reiten translation of cluster categories induces an algebra automorphism of $\mathcal {D}\mathcal {H}_\Lambda^{{cl}}(\widetilde{\mathcal{A}})$. Finally, we apply the BGP-reflections to $\mathcal {D}\mathcal {H}_\Lambda^{{cl}}(\widetilde{\mathcal{A}})$, and prove that the Hall algebra $\mathcal {D}\mathcal {H}_\Lambda^{{cl}}(\widetilde{\mathcal{A}})$ is invariant up to isomorphism under the mutations at sink vertices of $Q$.

Let $(X,r)$ be a distributive set-theoretical solution of the Yang-Baxter equation and $\mathcal{A}(X,r)$ the associated Yang-Baxter algebra. We prove that $\mathcal{A}(X,r)$ is isomorphic to a skew polynomial algebra and compute its Nakayama automorphism explicitly. We study the action of the permutation group $\mathcal{G}(X,r)$. It induces a subgroup $\overline{\mathcal{G}}\subseteq\operatorname{Aut}(\mathcal{A}(X,r))$, the induced automorphism group, for which $\mathcal{A}(X,r)$ is a faithful module. We characterize when $\overline{\mathcal{G}}$ is a reflection group and, in that case, describe the invariant subalgebra $\mathcal{A}(X,r)^{\overline{\mathcal{G}}}$ together with its Jacobian, reflection arrangement and discriminant. We further establish the Auslander theorem for a large class of distributive Yang-Baxter algebras. Finally, for a class of nontrivial distributive Yang–Baxter algebras, we obtain a lower bound for the pertinency of the group action induced by the permutation group.

Let $G$ be a standard reductive group over an algebraically closed field of positive characteristic and let $\mathfrak{g} = \operatorname{Lie}(G)$. The category of Harish-Chandra bimodules consists of $G$-equivariant $U(\mathfrak{g})$-bimodules such that the adjoint $\mathfrak{g}$-action integrates to a $G$-action. In this paper we classify the simple Harish-Chandra bimodules and determine their central characters.

Let $R$ be non-commutative ring with unity. In this paper, we prove that a given matrix in $M_n(R)$ is a sum of $p$-th powers in $M_n(R)$, where $n < p$ if and only if its trace can be written as a sum of $p$-th powers in $M_n(R)$, for $n = 2,3,4$ and $p = 5$ and $n = 4,5,6$ for $p = 7$. This paper extends the results of S. A. Katre and Kshipra Wadikar, $(p=3)$ and Garge $(n \geq p), (p = 5, 7)$, and S. A. Katre and Deepa Krishnamurthi $(n \geq p)$.

We present a general study of extending structures in the linear framework of Universal Algebra, which unifies many existing results and contains applications to various Poisson type algebras including transposed Poisson algebras.