Simple-minded Systems Research Articles

On simple-minded systems over domestic Brauer graph algebras

On simple-minded systems over domestic Brauer graph algebras

Reductions of triangulated categories and simple‐minded collections

AbstractSilting and Calabi–Yau reductions are important processes in representation theory to construct new triangulated categories from given ones, which are similar to Verdier quotient. In this paper, first we introduce a new reduction process of triangulated category, which is analogous to the silting (Calabi–Yau) reduction. For a triangulated category with a pre‐simple‐minded collection (pre‐SMC) , we construct a new triangulated category such that the SMCs in bijectively correspond to those in containing . Second, we give an analogue of Buchweitz's theorem for the singularity category of a SMC quadruple : the category can be realized as the stable category of an extriangulated subcategory of . Finally, we show the simple‐minded system (SMS) reduction due to Coelho Simões and Pauksztello is the shadow of our SMC reduction. This is parallel to the result that Calabi–Yau reduction is the shadow of silting reduction due to Iyama and Yang.

Positive Fuss–Catalan Numbers and Simple-Minded Systems in Negative Calabi–Yau Categories

Abstract We establish a bijection between $d$-simple-minded systems ($d$-SMSs) of $(-d)$-Calabi–Yau cluster category $\mathcal C_{-d}(H)$ and silting objects of ${\mathcal {D}}^{\mathrm {b}}(H)$ contained in ${\mathcal {D}}^{\le 0}\cap {\mathcal {D}}^{\ge 1-d}$ for hereditary algebra $H$ of Dynkin type and $d\ge 1$. We show that the number of $d$-SMSs in $\mathcal C_{-d}(H)$ is the positive Fuss–Catalan number $C_{d}^{+}(W)$ of the corresponding Weyl group $W$, by applying this bijection and Buan–Reiten–Thomas’ and Zhu’s results on Fomin–Reading’s generalized cluster complexes. Our results are based on a refined version of silting-$t$-structure correspondence.

Functorially finite hearts, simple-minded systems in negative cluster categories, and noncrossing partitions

Let $Q$ be an acyclic quiver and $w \geqslant 1$ be an integer. Let $\mathsf {C}_{-w}({\mathbf {k}} Q)$ be the $(-w)$-cluster category of ${\mathbf {k}} Q$. We show that there is a bijection between simple-minded collections in $\mathsf {D}^b({\mathbf {k}} Q)$ lying in a fundamental domain of $\mathsf {C}_{-w}({\mathbf {k}} Q)$ and $w$-simple-minded systems in $\mathsf {C}_{-w}({\mathbf {k}} Q)$. This generalises the same result of Iyama–Jin in the case that $Q$ is Dynkin. A key step in our proof is the observation that the heart $\mathsf {H}$ of a bounded t-structure in a Hom-finite, Krull–Schmidt, ${\mathbf {k}}$-linear saturated triangulated category $\mathsf {D}$ is functorially finite in $\mathsf {D}$ if and only if $\mathsf {H}$ has enough injectives and enough projectives. We then establish a bijection between $w$-simple-minded systems in $\mathsf {C}_{-w}({\mathbf {k}} Q)$ and positive $w$-noncrossing partitions of the corresponding Weyl group $W_Q$.

On Simple-Minded Systems Over Representation-Finite Self-Injective Algebras

Let A be a representation-finite self-injective algebra over an algebraically closed field k. We give a new characterization for an orthogonal system in the stable module category A-\(\underline {\text {mod}}\) to be a simple-minded system. As a by-product, we show that every Nakayama-stable orthogonal system in A-\(\underline {\text {mod}}\) extends to a simple-minded system.

An explicit construction of simple-minded systems over self-injective Nakayama algebras

Recently, we obtained a new characterization for an orthogonal system to be a simple-minded system in the stable module category of any representation-finite self-injective algebra. In this paper, we apply this result to give an explicit construction of s

On simple-minded systems and τ-periodic modules of self-injective algebras

Let A be a finite-dimensional self-injective algebra over an algebraically closed field, C a stably quasi-serial component (i.e. its stable part is a tube) of rank n of the Auslander-Reiten quiver of A, and S be a simple-minded system of the stable module category mod_A. We show that the intersection S∩C is of size strictly less than n, and consists only of modules with quasi-length strictly less than n. In particular, all modules in the homogeneous tubes of the Auslander-Reiten quiver of A cannot be in any simple-minded system.

Simple-minded systems and reduction for negative Calabi-Yau triangulated categories

We develop the basic properties of w w -simple-minded systems in ( − w ) (-w) -Calabi-Yau triangulated categories for w ⩾ 1 w \geqslant 1 . We show that the theory of simple-minded systems exhibits striking parallels with that of cluster-tilting objects. The main result is a reduction technique for negative Calabi-Yau triangulated categories. Our construction provides an inductive technique for constructing simple-minded systems.

Multiband model for the electronic structure of Sr2TiO4

AbstractWe introduce and investigate the multiband d–p model for a TiO layer, such as realized in SrTiO, with all and orbitals (at titanium ions) and orbitals (at oxygen ions). Complementary density functional theory ab initio computations are employed to determine the actual electron number per TiO unit and one finds perfect Sr ionization with Sr ions and charged (TiO) layer. This system is predicted to be a robust nonmagnetic insulator, in agreement with experiment. The above charge distribution is crucial and when we deviate from it, even by a small amount, the system becomes conducting or very close to conducting and various magnetic structures compete with one another. This finding is generic, that is, it holds in a broad range of d–p Hamiltonian parameters. As expected, d–p hybridization strongly redistributes electrons and leads to titanium ions between and ionic configurations. Surprisingly, SrTiO is not a simpleminded system but instead electron densities are finite and roughly the same in all different orbitals (of and symmetry) and the electron densities within O() orbitals are within the range 5.4–5.9. By selecting the charge‐transfer energy, , we reproduce the experimental band gap of 3.8 eV. We emphasize that a realistic treatment of electronic distribution requires local Coulomb interactions at oxygen orbitals and we show that the Coulomb interactions at titanium ions are strongly renormalized when the Coulomb interactions at oxygen ions are neglected.

Mutations of simple-minded systems in Calabi–Yau categories generated by a spherical object

Abstract In this article, we give a definition and a classification of ‘higher’ simple-minded systems in triangulated categories generated by spherical objects with negative Calabi–Yau dimension. We also study mutations of this class of objects and that of ‘higher’ Hom-configurations and Riedtmann configurations. This gives an explicit analogue of the ‘nice’ mutation theory exhibited in cluster-tilting theory.

Two-term Tilting Complexes and Simple-minded Systems of Self-injective Nakayama Algebras

We study the relation between simple-minded systems and two-term tilting complexes for self-injective Nakayama algebras. More precisely, we show that any simple-minded system of a self-injective Nakayama algebra is the image of the set of simple modules under a stable equivalence, which is given by the restriction of a standard derived equivalence induced by a two-term tilting complex. We achieve this by exploiting and connecting the mutation theories from the combinatorics of Brauer tree, configurations of stable translations quivers of type A, and triangulations of a punctured convex regular polygon.

Simple-minded systems, configurations and mutations for representation-finite self-injective algebras

Simple-minded systems of objects in a stable module category are defined by common properties with the set of simple modules, whose images under stable equivalences do form simple-minded systems. Over a representation-finite self-injective algebra, it is shown that all simple-minded systems are images of simple modules under stable equivalences of Morita type, and that all simple-minded systems can be lifted to Nakayama-stable simple-minded collections in the derived category. In particular, all simple-minded systems can be obtained algorithmically using mutations.

Torsion Pairs and Simple-Minded Systems in Triangulated Categories

Let T be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu, we say that a family S of pairwise orthogonal objects in T with trivial endomorphism rings is a simple-minded system if its closure under extensions is all of T. We construct torsion pairs in T associated to any subset X of a simple-minded system S, and use these to define left and right mutations of S relative to X. When T has a Serre functor \nu, and S and X are invariant under \nu[1], we show that these mutations are again simple-minded systems. We are particularly interested in the case where T is the stable module category of a self-injective algebra \Lambda. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in the derived category of \Lambda. It follows that the mutation of the set of simple \Lambda-modules relative to X yields the images of the simple \Gamma-modules under a stable equivalence between \Gamma\ and \Lambda, where \Gamma\ is the tilting mutation of \Lambda\ relative to X.

SIMPLE-MINDED SYSTEMS IN STABLE MODULE CATEGORIES

Simple-minded systems in stable module categories are defined by orthogonality and generating properties so that the images of the simple modules under a stable equivalence form such a system. Simple-minded systems are shown to be invariant under stable equivalences; thus the set of all simple-minded systems is an invariant of a stable module category. The simple-minded systems of several classes of algebras are described and connections to the Auslander-Reiten conjecture are pointed out.

The circulatory and metabolic correlates of functional activity

Simple-minded systems of objects in a stable module category are defined by common properties with the set of simple modules, whose images under stable equivalences do form simple-minded systems. Over a representation-finite self-injective algebra, it is shown that all simple-minded systems are images of simple modules under stable equivalences of Morita type, and that all simple-minded systems can be lifted to Nakayama-stable simple-minded collections in the derived category. In particular, all simple-minded systems can be obtained algorithmically using mutations.

DEVELOPMENTAL AND MOTIVATIONAL CONCEPTS IN PIAGET'S SENSORIMOTOR THEORY OF INTELLIGENCE

Medical materialism seems indeed a good appellation for the too simple-minded system of thought which we are considering. Medical materialism finishes up Saint Paul by calling his vision on the road to Damascus a discharging lesion of the occipital cortex, he heing an epileptic. It snuffs out Saint Theresa as an hysteric, Sail lt Francis of Assisi as an hereditary degenerate.... Carlyle's org, -tones of misery it accounts for by a gastro-duodenal catarrh [po 14]. A more fully developed example of the same kind ofreasoning is the fashion, quite common nowadays among certain writers, of <.Titicizing the religious emotions by showing a connection between them and the sexual life. Conversion is a crisis of puberty and adolescence. The macerations of saints, and the devotion of missionaries, are only instances of the parental instinct of selfsacrifice gone astray. For the hysterical nun, starving for natural life, Christ is but an imaginary substitute for a more earthly subject of affection [po 11£.].