In this paper, we show that for a large natural class of vertex operator algebras (VOAs) and their modules, the Zhu’s algebras and bimodules (and their g-twisted analogs) are Noetherian. These carry important information about the representation theory of the VOA, and its fusion rules, and the Noetherian property gives the potential for (non-commutative) algebro-geometric methods to be employed in their study.

A complete description of all possible multiplicative groups of finite skew left braces whose additive group has trivial centre is given. As a consequence, some earlier results of Tsang can be improved and an answer to an open question set by Tsang at Ischia Group Theory 2024 Conference is provided.

Let R be a ring and let $n\ge 2$. We discuss the question of whether every element in the matrix ring $M_n(R)$ is a product of (additive) commutators $[x,y]=xy-yx$, for $x,y\in M_n(R)$. An example showing that this does not always hold, even when R is commutative, is provided. If, however, R has Bass stable rank one, then under various additional conditions every element in $M_n(R)$ is a product of three commutators. Further, if R is a division ring with infinite center, then every element in $M_n(R)$ is a product of two commutators. If R is a field and $a\in M_n(R)$, then every element in $M_n(R)$ is a sum of elements of the form $[a,x][a,y]$ with $x,y\in M_n(R)$ if and only if the degree of the minimal polynomial of a is greater than $2$.

In this paper, we investigate locally finitely presented pure semisimple (hereditary) Grothendieck categories. We show that every locally finitely presented pure semisimple (resp., hereditary) Grothendieck category $\mathscr {A}$ is equivalent to the category of left modules over a left pure semisimple (resp., left hereditary) ring when $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ is a QF-3 category, and every representable functor in $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ has finitely generated essential socle. In fact, we show that there exists a bijection between Morita equivalence classes of left pure semisimple (resp., left hereditary) rings $\Lambda $ and equivalence classes of locally finitely presented pure semisimple (resp., hereditary) Grothendieck categories $\mathscr {A}$ that $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ is a QF-3 category, and every representable functor in $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ has finitely generated essential socle. To prove this result, we study left pure semisimple rings by using Auslander’s ideas. We show that there exists, up to equivalence, a bijection between the class of left pure semisimple rings and the class of rings with nice homological properties. These results extend the Auslander and Ringel–Tachikawa correspondence to the class of left pure semisimple rings. As a consequence, we give several equivalent statements to the pure semisimplicity conjecture.

We present a construction of left braces of right nilpotency class at most two based on suitable actions of an abelian group on itself with an invariance condition. This construction allows us to recover the construction of a free right nilpotent one-generated left brace of class two.

The proof of Theorem 3.14 contains an unsubstantiated claim. To overcome this problem, we add a hypothesis to the statement of 3.14 and we provide a new valid proof. We adjust Theorem 3.15, Corollary 3.16, Proposition 4.23, Theorem 4.26, Corollary 4.29, and Corollary 4.32 accordingly.

Nilpotency concepts for skew braces are among the main tools with which we are nowadays classifying certain special solutions of the Yang–Baxter equation, a consistency equation that plays a relevant role in quantum statistical mechanics and in many areas of mathematics. In this context, two relevant questions have been raised in F. Cedó, A. Smoktunowicz and L. Vendramin (Skew left braces of nilpotent type. Proc. Lond. Math. Soc. (3) 118 (2019), 1367–1392) (see questions 2.34 and 2.35) concerning right- and central nilpotency. The aim of this short note is to give a negative answer to both questions: thus, we show that a finite strong-nil brace B need not be right-nilpotent. On a positive note, we show that there is one (and only one, by our examples) special case of the previous questions that actually holds. In fact, we show that if B is a skew brace of nilpotent type and $b\ \ast \ b=0$ for all $b\in B$, then B is centrally nilpotent.

We establish a McKay correspondence for finite and linearly reductive subgroup schemes of ${\mathbf {SL}}_2$ in positive characteristic. As an application, we obtain a McKay correspondence for all rational double point singularities in characteristic $p\geq 7$. We discuss linearly reductive quotient singularities and canonical lifts over the ring of Witt vectors. In dimension 2, we establish simultaneous resolutions of singularities of these canonical lifts via G-Hilbert schemes. In the appendix, we discuss several approaches towards the notion of conjugacy classes for finite group schemes: This is an ingredient in McKay correspondences, but also of independent interest.

In this paper, we develop two new homological invariants called relative dominant dimension with respect to a module and relative codominant dimension with respect to a module. Among the applications are precise connections between Ringel duality, split quasi-hereditary covers and double centralizer properties, constructions of split quasi-hereditary covers of quotients of Iwahori-Hecke algebras using Ringel duality of q-Schur algebras and a new proof for Ringel self-duality of the blocks of the Bernstein-Gelfand-Gelfand category $\mathcal {O}$. These homological invariants are studied over Noetherian algebras which are finitely generated and projective as a module over the ground ring. They are shown to behave nicely under change of rings techniques.

Given any smooth germ of a 3-fold flopping contraction, we first give a combinatorial characterisation of which Gopakumar–Vafa (GV) invariants are non-zero, by prescribing multiplicities to the walls in the movable cone. On the Gromov–Witten (GW) side, this allows us to describe, and even draw, the critical locus of the associated quantum potential. We prove that the critical locus is the infinite hyperplane arrangement of Iyama and the second author and, moreover, that the quantum potential can be reconstructed from a finite fundamental domain. We then iterate, obtaining a combinatorial description of the matrix that controls the transformation of the non-zero GV invariants under a flop. There are three main ingredients and applications: (1) a construction of flops from simultaneous resolution via cosets, which describes how the dual graph changes; (2) a closed formula, which describes the change in dimension of the contraction algebra under flop; and (3) a direct and explicit isomorphism between quantum cohomologies of different crepant resolutions, giving a Coxeter-style, visual proof of the Crepant Transformation Conjecture for isolated cDV singularities.

We study Morita equivalence for idempotent rings with involution. Following the ideas of Rieffel, we define Rieffel contexts, and we also introduce Morita $*$-contexts and enlargements for rings with involution. We prove that two idempotent rings with involution have a joint enlargement if and only if they are connected by a unitary and full Rieffel context. These conditions are also equivalent to having a unitary and surjective Morita $*$-context between those rings. We also examine how the mentioned conditions are connected to the existence of certain equivalence functors between the categories of firm modules over the given rings with involution.

Given a non-negative integer n and a ring R with identity, we construct a hereditary abelian model structure on the category of left R-modules where the class of cofibrant objects coincides with $\mathcal{GF}_n(R)$ the class of left R-modules with Gorenstein flat dimension at most n, the class of fibrant objects coincides with $\mathcal{F}_n(R)^\perp$ the right ${\rm Ext}$-orthogonal class of left R-modules with flat dimension at most n, and the class of trivial objects coincides with $\mathcal{PGF}(R)^\perp$ the right ${\rm Ext}$-orthogonal class of PGF left R-modules recently introduced by Šaroch and . The homotopy category of this model structure is triangulated equivalent to the stable category $\underline{\mathcal{GF}(R)\cap\mathcal{C}(R)}$ modulo flat-cotorsion modules and it is compactly generated when R has finite global Gorenstein projective dimension.

The second part of this paper deals with the PGF dimension of modules and rings. Our results suggest that this dimension could serve as an alternative definition of the Gorenstein projective dimension. We show, among other things, that (n-)perfect rings can be characterized in terms of Gorenstein homological dimensions, similar to the classical ones, and the global Gorenstein projective dimension coincides with the global PGF dimension.

Previously [‘Radicals and idempotents I, II’, Comm. Alg. 49(1) (2021), 73–84 and 50(11) (2022), 4791–4804], we have studied the interaction between radicals of rings and idempotents in general or those of particular types, for example, left semicentral. Here we carry out similar investigations for q-central idempotents, that is, those idempotents e satisfying the condition $(ea-eae)(be-ebe) = 0$ for all a, b.

We introduce the concept of ‘irrational paths’ for a given subshift and useit to characterize all minimal left ideals in the associated unital subshift algebra. Consequently, we characterize the socle as the sum of the ideals generated by irrational paths. Proceeding, we construct a graph such that the Leavitt path algebra of this graph is graded isomorphic to the socle. This realization allows us to show that the graded structure of the socle serves as an invariant for the conjugacy of Ott–Tomforde–Willis subshifts and for the isometric conjugacy of subshifts constructed with the product topology. Additionally, we establish that the socle of the unital subshift algebra is contained in the socle of the corresponding unital subshift C*-algebra.

We propose a notion of multi-scale stability conditions with the goal of providing a smooth compactification of the quotient of the space of projectivized Bridgeland stability conditions by the group of autoequivalence. For the case of the 3CY category associated with the $A_n$-quiver, this goal is achieved by defining a topology and complex structure that relies on a plumbing construction.

We compare this compactification to the multi-scale compactification of quadratic differentials and briefly indicate why even for the Kronecker quiver, this notion needs refinement to provide a full compactification.

We introduce non-associative skew Laurent polynomial rings and characterize when they are simple. Thereby, we generalize results by Jordan, Voskoglou, and Nystedt and Öinert.

In this article, we study rational matrix representations of VZ p-groups (p is any prime). Using our findings on VZ p-groups, we explicitly obtain all inequivalent irreducible rational matrix representations of all p-groups of order $\leq p^4$. Furthermore, we establish combinatorial formulae to determine the Wedderburn decompositions of rational group algebras for VZ p-groups and all p-groups of order $\leq p^4$, ensuring simplicity in the process.

Let A be an F-central simple algebra of degree $m^2=\prod _{i=1}^k p_i^{2\alpha _i}$ and G be a subgroup of the unit group of A such that $F[G]=A$. We prove that if G is a central product of two of its subgroups M and N, then $F[M]\otimes _F F[N]\cong F[G]$. Also, if G is locally nilpotent, then G is a central product of subgroups $H_i$, where $[F[H_i]:F]=p_i^{2\alpha _i}$, $A=F[G]\cong F[H_1]\otimes _F \cdots \otimes _F F[H_k]$ and $H_i/Z(G)$ is the Sylow $p_i$-subgroup of $G/Z(G)$ for each i with $1\leq i\leq k$. Additionally, there is an element of order $p_i$ in F for each i with $1\leq i\leq k$.

Tachikawa's second conjecture for symmetric algebras is shown to be equivalent to indecomposable symmetric algebras not having any nontrivial stratifying ideals. The conjecture is also shown to be equivalent to the supremum of stratified ratios being less than $1$, when taken over all indecomposable symmetric algebras. An explicit construction provides a series of counterexamples to Tachikawa's second conjecture from each (potentially existing) gendo-symmetric algebra that is a counterexample to Nakayama's conjecture. The results are based on establishing recollements of derived categories and on constructing new series of algebras.

Dualities of resolving subcategories of module categories over rings are introduced and characterized as dualities with respect to Wakamatsu tilting bimodules. By restriction of the dualities to smaller resolving subcategories, sufficient and necessary conditions for these bimodules to be tilting are provided. This leads to the Gorenstein version of both the Miyashita’s duality and Huisgen-Zimmermann’s correspondence. An application of resolving dualities is to show that higher algebraic K-groups and semi-derived Ringel–Hall algebras of finitely generated Gorenstein-projective modules over Artin algebras are preserved under tilting.