Hall algebra of morphism category

QingHua Chen; Liwang Zhang

Displaying similar documents to “Hall algebra of morphism category”

SCAP-subalgebras of Lie algebras

Sara Chehrazi, Ali Reza Salemkar (2016)

Czechoslovak Mathematical Journal

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A subalgebra $H$ of a finite dimensional Lie algebra $L$ is said to be a $SCAP$ -subalgebra if there is a chief series $0 = L_{0} \subset L_{1} \subset ... \subset L_{t} = L$ of $L$ such that for every $i = 1, 2, ..., t$ , we have $H + L_{i} = H + L_{i - 1}$ or $H \cap L_{i} = H \cap L_{i - 1}$ . This is analogous to the concept of $SCAP$ -subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its $SCAP$ -subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable.

Hall algebras of two equivalent extriangulated categories

Shiquan Ruan, Li Wang, Haicheng Zhang (2024)

Czechoslovak Mathematical Journal

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For any positive integer $n$ , let $A_{n}$ be a linearly oriented quiver of type $A$ with $n$ vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories $ℳ_{n + 1}$ and $ℱ_{n}$ , where $ℳ_{n + 1}$ and $ℱ_{n}$ are the two extriangulated categories corresponding to the representation category of $A_{n + 1}$ and the morphism category of projective representations of $A_{n}$ , respectively. As a...

On the structure of triangulated categories with finitely many indecomposables

Claire Amiot (2007)

Bulletin de la Société Mathématique de France

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We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field $k$ . We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category $𝒯$ is of the form $ℤ Δ / G$ where $Δ$ is a disjoint union of simply-laced Dynkin diagrams and $G$ a weakly admissible group of automorphisms of $ℤ Δ$ . Then we prove that for ‘most’ groups $G$ , the category $𝒯$ is standard, ...

On category $𝒪$ for cyclotomic rational Cherednik algebras

Iain G. Gordon, Ivan Losev (2014)

Journal of the European Mathematical Society

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We study equivalences for category $𝒪_{p}$ of the rational Cherednik algebras $𝐇_{p}$ of type $G_{ℓ} (n) = {(μ_{ℓ})}^{n} ⋊ 𝔖_{n}$ : a highest weight equivalence between $𝒪_{p}$ and $𝒪_{σ (p)}$ for $σ \in 𝔖_{ℓ}$ and an action of $𝔖_{ℓ}$ on an explicit non-empty Zariski open set of parameters $p$ ; a derived equivalence between $𝒪_{p}$ and $𝒪_{p^{'}}$ whenever $p$ and $p^{'}$ have integral difference; a highest weight equivalence between $𝒪_{p}$ and a parabolic category $𝒪$ for the general linear group, under a non-rationality assumption on the parameter $p$ . As a consequence, we confirm special cases...

Bipartite coalgebras and a reduction functor for coradical square complete coalgebras

Justyna Kosakowska, Daniel Simson (2008)

Colloquium Mathematicae

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Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C₁, the second term of the coradical filtration of C. If C = C₁, we associate with C a K-linear functor $ℍ_{C} : C - C o m o d \to H_{C} - C o m o d$ that restricts to a representation equivalence $ℍ_{C} : C - c o m o d \to H_{C} - c o m o d_{s p}^{•}$ , where $H_{C}$ is a coradical square complete hereditary...

Higher-dimensional Auslander-Reiten sequences

Jiangsha Li, Jing He (2024)

Czechoslovak Mathematical Journal

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Zhou and Zhu have shown that if $𝒞$ is an $(n + 2)$ -angulated category and $𝒳$ is a cluster tilting subcategory of $𝒞$ , then the quotient category $𝒞 / 𝒳$ is an $n$ -abelian category. We show that if $𝒞$ has Auslander-Reiten $(n + 2)$ -angles, then $𝒞 / 𝒳$ has Auslander-Reiten $n$ -exact sequences.

Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective

Xinfeng Liang, Feng Wei, Ajda Fošner (2019)

Czechoslovak Mathematical Journal

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Let $ℛ$ be a commutative ring, $𝒢$ be a generalized matrix algebra over $ℛ$ with weakly loyal bimodule and $𝒵 (𝒢)$ be the center of $𝒢$ . Suppose that $𝔮 : 𝒢 \times 𝒢 \to 𝒢$ is an $ℛ$ -bilinear mapping and that $𝔗_{𝔮} : 𝒢 \to 𝒢$ is a trace of $𝔮$ . The aim of this article is to describe the form of $𝔗_{𝔮}$ satisfying the centralizing condition $[𝔗_{𝔮} (x), x] \in 𝒵 (𝒢)$ (and commuting condition $[𝔗_{𝔮} (x), x] = 0$ ) for all $x \in 𝒢$ . More precisely, we will revisit the question of when the centralizing trace (and commuting trace) $𝔗_{𝔮}$ has the so-called proper form from a new perspective. Using the aforementioned...

Stable tubes in extriangulated categories

Li Wang, Jiaqun Wei, Haicheng Zhang (2022)

Czechoslovak Mathematical Journal

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Let $𝒳$ be a semibrick in an extriangulated category. If $𝒳$ is a $τ$ -semibrick, then the Auslander-Reiten quiver $Γ (ℱ (𝒳))$ of the filtration subcategory $ℱ (𝒳)$ generated by $𝒳$ is $ℤ 𝔸_{\infty}$ . If $𝒳 = {X_{i}}_{i = 1}^{t}$ is a $τ$ -cycle semibrick, then $Γ (ℱ (𝒳))$ is $ℤ 𝔸_{\infty} / τ_{𝔸}^{t}$ .

Limits and colimits in certain categories of spaces of continuous functions

Marvin W. Grossman

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CONTENTSIntroduction................................................................................................................................................................................5§ 1. Notation and preliminaries.............................................................................................................................................6§ 2. Epimorphisms and monomorphisms.........................................................................................................................7§...

$\mathcal{L}$ -homomorphisms of Lie algebras

Aleksander A. Lashkhi (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si studiano gli omomorfismi reticolari ( $\mathcal{L}$ -omomorfismi) di algebre di Lie sopra anelli commutativi con unità. Le algebre di Lie sopra un campo e le $p$ -algebre di Lie non ammettono $\mathcal{L}$ -omomorfismi propri. Si assegnano condizioni necessarie e sufficienti affinchè un'algebra di Lie periodica o mista possieda un « $\mathcal{L}$ -omomorfismo su una catena di lunghezza $n$ .

Multiplicative Lie triple derivations on standard operator algebras

Bilal Ahmad Wani (2021)

Communications in Mathematics

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Let $𝒳$ be a Banach space of dimension $n > 1$ and $𝔄 \subset ℬ (𝒳)$ be a standard operator algebra. In the present paper it is shown that if a mapping $d : 𝔄 \to 𝔄$ (not necessarily linear) satisfies $d ([[U, V], W]) = [[d (U), V], W] + [[U, d (V)], W] + [[U, V], d (W)]$ for all $U, V, W \in 𝔄$ , then $d = ψ + τ$ , where $ψ$ is an additive derivation of $𝔄$ and $τ : 𝔄 \to 𝔽 I$ vanishes at second commutator $[[U, V], W]$ for all $U, V, W \in 𝔄$ . Moreover, if $d$ is linear and satisfies the above relation, then there exists an operator $S \in 𝔄$ and a linear mapping $τ$ from $𝔄$ into $𝔽 I$ satisfying $τ ([[U, V], W]) = 0$ for all $U, V, W \in 𝔄$ , such that $d (U) = S U - U S + τ (U)$ for all $U \in 𝔄$ .

Cotorsion pairs in comma categories

Yuan Yuan, Jian He, Dejun Wu (2024)

Czechoslovak Mathematical Journal

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Let $𝒜$ and $ℬ$ be abelian categories with enough projective and injective objects, and $T : 𝒜 \to ℬ$ a left exact additive functor. Then one has a comma category $(ℬ ↓ T)$ . It is shown that if $T : 𝒜 \to ℬ$ is $𝒳$ -exact, then $(^{⊥} 𝒳, 𝒳)$ is a (hereditary) cotorsion pair in $𝒜$ and $(^{⊥} 𝒴, 𝒴)$ ) is a (hereditary) cotorsion pair in $ℬ$ if and only if $((\binom{^{⊥} 𝒳}{^{⊥} 𝒴})), 〈 𝐡 (𝒳, 𝒴) 〉)$ is a (hereditary) cotorsion pair in $(ℬ ↓ T)$ and $𝒳$ and $𝒴$ are closed under extensions. Furthermore, we characterize when special preenveloping classes in abelian categories $𝒜$ and $ℬ$ can induce special preenveloping...

$𝔤$ -quasi-Frobenius Lie algebras

David N. Pham (2016)

Archivum Mathematicum

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A Lie version of Turaev’s $\bar{G}$ -Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a $𝔤$ -quasi-Frobenius Lie algebra for $𝔤$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(𝔮, β)$ together with a left $𝔤$ -module structure which acts on $𝔮$ via derivations and for which $β$ is $𝔤$ -invariant. Geometrically, $𝔤$ -quasi-Frobenius Lie algebras are the Lie algebra structures associated to...

On almost complex structures from classical linear connections

Jan Kurek, Włodzimierz M. Mikulski (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $ℳ f_{m}$ be the category of $m$ -dimensional manifolds and local diffeomorphisms and let $T$ be the tangent functor on $ℳ f_{m}$ . Let $𝒱$ be the category of real vector spaces and linear maps and let $𝒱_{m}$ be the category of $m$ -dimensional real vector spaces and linear isomorphisms. We characterize all regular covariant functors $F : 𝒱_{m} \to 𝒱$ admitting $ℳ f_{m}$ -natural operators $\tilde{J}$ transforming classical linear connections $\nabla$ on $m$ -dimensional manifolds $M$ into almost complex structures $\tilde{J} (\nabla)$ on $F (T) M = ⋃_{x \in M} F (T_{x} M)$ .

On the nilpotent residuals of all subalgebras of Lie algebras

Wei Meng, Hailou Yao (2018)

Czechoslovak Mathematical Journal

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Let $𝒩$ denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra $L$ over an arbitrary field $𝔽$ , there exists a smallest ideal $I$ of $L$ such that $L / I \in 𝒩$ . This uniquely determined ideal of $L$ is called the nilpotent residual of $L$ and is denoted by $L^{𝒩}$ . In this paper, we define the subalgebra $S (L) = ⋂_{H \leq L} I_{L} (H^{𝒩})$ . Set $S_{0} (L) = 0$ . Define $S_{i + 1} (L) / S_{i} (L) = S (L / S_{i} (L))$ for $i \geq 1$ . By $S_{\infty} (L)$ denote the terminal term of the ascending series. It is proved that $L = S_{\infty} (L)$ if and only if $L^{𝒩}$ is nilpotent. In addition, we investigate the basic properties of a...