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We determine the number of isomorphism classes of elementary gradings by a finite group on an algebra of upper block-triangular matrices. As a consequence we prove that, for a finite abelian group $G$, the sequence of the numbers $E(G,m)$…

Octupolar order is described in two space dimensions in terms of the maxima (and conjugated minima) of the probability density associated with a third-rank, fully symmetric and traceless tensor. Such a representation is shown to be…

In this article we introduce and study a class of finite groups for which the orders of normal subgroups satisfy a certain inequality. It is closely connected to some well-known arithmetic classes of natural numbers.

We prove a stronger version of the octahedral axiom in a pre-triangulated category. The proof uses a new lemma about exact sequences in pointed additive categories which is based on a weak converse of the snake lemma.

The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of…

We compute the first extension group from a simple object to a proper standard object and, in some cases, the first extension group from a simple object to a standard object in the principal block of an $\mathcal{S}$-subcategory of the BGG…

Given a thick subcategory of a triangulated category, we define a colocalisation and a natural long exact sequence that involves the original category and its localisation and colocalisation at the subcategory. Similarly, we construct a…

In this paper, we explore when a locally finite triangulated category has dimension zero or finite representation type. We also study generation of derived categories by orthogonal subcategories.

For a countable, complete, first-order theory $T$, we study $At$, the class of atomic models of $T$. We develop an analogue of $U$-rank and prove two results. On one hand, if some tp(d/a) is not ranked, then there are $2^{\aleph_1}$…

We introduce several notions and give examples. We prove that ${\rm Stab}(D^b(K(l)))\cong {\mathbb C}\times \mathcal H$ for $l\geq 3$, where $K(l)$ is $l$-Kronecker quiver. This is an example of SOD, where ${\rm Stab}( \langle \mathcal…

The main result of this paper is that there is an additive equivalence between $\overline{\mathcal{C}}_n$, the Paquette-Yildirim completion of the discrete cluster categories of Dynkin type $A_{\infty}$, and the perfect derived category of…

A computable structure $\mathcal{A}$ has degree of categoricity $\mathbf{d}$ if $\mathbf{d}$ is exactly the degree of difficulty of computing isomorphisms between isomorphic computable copies of $\mathcal{A}$. Fokina, Kalimullin, and Miller…

We define a notion of categorical first order deformations for (enhanced) triangulated categories. For a category $\mathcal{T}$, we show that there is a bijection between $\operatorname{HH}^2(\mathcal{T})$ and the set of categorical…

Any simple group-grading of a finite dimensional complex algebra induces a natural family of digraphs. We prove that $|E\circ E^{\text{op}}\cup E^{\text{op}}\circ E|\geq |E|$ for any digraph $\Gamma =(V,E)$ without parallel edges, and…

Let $A$ be a finite dimensional $G$-graded algebra with $G$ a finite group, and $A\# k[G]^{\ast}$ be the smash product of $A$ with the group $G$. Our results can be stated as follows: (1) If $A$ is a self-injective algebra and separably…

We introduce the notions of a $\mathbf{D}$-standard abelian category and a $\mathbf{K}$-standard additive category. We prove that for a finite dimensional algebra $A$, its module category is $\mathbf{D}$-standard if and only if any derived…

We give a detailed proof of the following fundamental result: the singularity category of a ring is triangle equivalent to the stabilization of its stable module category. The result yields singular equivalences between rings of different…

In this paper, we consider semi-extraspecial $p$-groups $G$ that have an automorphism of order $|G:G'| - 1$. We prove that these groups are isomorphic to Sylow $p$-subgroups of ${\rm SU}_3 (p^{2a})$ for some integer $a$. If $p$ is odd, this…

We show that the derived category of any singularity over a field of characteristic 0 can be embedded fully and faithfully into a smooth triangulated category which has a semiorthogonal decomposition with components equivalent to derived…

In this paper we explain certain systematic differences between algebraic and topological triangulated categories. A triangulated category is algebraic if it admits a differential graded model, and topological if it admits a model in the…