Journal of Algebra最新文献

英文中文

Extending structures for dendriform algebras 树枝状代数的扩展结构

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-15 DOI: 10.1016/j.jalgebra.2024.09.025

Yuanyuan Zhang, Junwen Wang

In this paper, we devote to extending structures for dendriform algebras. First, we define extending datums and unified products of dendriform algebras, and theoretically solve the extending structure problem. As an application, we consider flag datums as a special case of extending structures, and give an example of the extending structure problem. Second, we apply matched pairs and bicrossed products of dendriform algebras and theoretically solve the factorization problem for dendriform algebras. Moreover, we also introduce cocycle semidirect products and nonabelian semidirect products as special cases of unified products. Finally, we define the deformation map on a dendriform extending structure (more general case), not necessary a matched pair, which is more practical in the classifying complements problem.

本文致力于树状结构的扩展结构。首先，我们定义了树形结构的扩展基准和统一积，并从理论上解决了扩展结构问题。作为一种应用，我们把标志基准视为扩展结构的一种特例，并给出了扩展结构问题的一个例子。其次，我们应用了树状结构的配对和双交积，并从理论上解决了树状结构的因式分解问题。此外，我们还介绍了作为统一积特例的循环半间接积和非阿贝尔半间接积。最后，我们定义了树枝状扩展结构上的变形映射（更一般的情况），它不一定是匹配对，这在分类补全问题中更实用。

引用次数: 0

Unramified Brauer group of quotient spaces by finite groups 有限群商数空间的非ramified Brauer 群

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-15 DOI: 10.1016/j.jalgebra.2024.10.006

Andrew Kresch , Yuri Tschinkel

We give a general procedure to determine the unramified Brauer group of quotients of rational varieties by finite groups.

我们给出了确定有限群有理子商的未成帧布劳尔群的一般程序。

引用次数: 0

Noncommutative differential geometry on crossed product algebras 交叉积代数上的非交换微分几何

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-15 DOI: 10.1016/j.jalgebra.2024.10.007

Andrea Sciandra , Thomas Weber

We provide a differential structure on arbitrary cleft extensions

B : = A^{co H} \subseteq A

for an H-comodule algebra A. This is achieved by constructing a covariant calculus on the corresponding crossed product algebra

B #_{σ} H

from the data of a bicovariant calculus on the structure Hopf algebra H and a calculus on the base algebra B, which is compatible with the 2-cocycle and measure of the crossed product. The result is a quantum principal bundle with canonical strong connection and we describe the induced bimodule covariant derivatives on associated bundles of the crossed product. All results specialize to trivial extensions and smash product algebras B#H and we give a characterization of the smash product calculus in terms of the differentials of the cleaving map

j : H \to A

and the inclusion

B ↪ A

. The construction is exemplified for pointed Hopf algebras. In particular, the case of Radford Hopf algebras

H_{(r, n, q)}

is spelled out in detail.

我们提供了一个关于任意劈裂扩展B:=AcoH⊆A的微分结构，它是由结构霍普夫代数H上的双变量微积分数据和基代数B上的微积分数据，通过在相应的交叉积代数B#σH上构造一个协变量微积分来实现的，这个协变量微积分与交叉积的2周期和度量相容。结果是一个具有典型强连接的量子主束，我们描述了交叉积相关束上的诱导双模协变导数。所有结果都专门用于琐碎扩展和粉碎积代数 B#H，我们还给出了粉碎积微积分在裂开映射 j:H→A 和包含 BA 的微分方面的特征。我们以尖霍普夫原子为例说明了这一构造。特别是详细说明了拉德福德霍普夫代数方程 H(r,n,q) 的情况。

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引用次数: 0

Invariants of r-spin TQFTs and non-semisimplicity r-自旋 TQFT 的不变式与非半简性

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-15 DOI: 10.1016/j.jalgebra.2024.08.039

Nils Carqueville , Ehud Meir , Lóránt Szegedy

For a positive integer r, an r-spin topological quantum field theory is a 2-dimensional TQFT with tangential structure given by the r-fold cover of

{SO}_{2}

. In particular, such a TQFT assigns a scalar invariant to every closed r-spin surface Σ. Given a sequence of scalars indexed by the set of diffeomorphism classes of all such Σ, we construct a symmetric monoidal category

C

and a

C

-valued r-spin TQFT which reproduces the given sequence. We also determine when such a sequence arises from a TQFT valued in an abelian category with finite-dimensional Hom spaces. In particular, we construct TQFTs with values in super vector spaces that can distinguish all diffeomorphism classes of r-spin surfaces, and we show that the Frobenius algebras associated to such TQFTs are necessarily non-semisimple.

对于正整数 r，r-旋拓扑量子场论是一种二维 TQFT，其切向结构由 SO2 的 r 叠盖给出。尤其是，这样的 TQFT 会给每个封闭的 r 自旋面 Σ 分配一个标量不变量。给定一个由所有这样的 Σ 的差分类集合索引的标量序列，我们将构造一个对称单环范畴 C 和一个重现给定序列的 C 值 r-自旋 TQFT。我们还确定了在具有有限维 Hom 空间的无性范畴中估值的 TQFT 何时会产生这样的序列。特别是，我们构建了在超向量空间中取值的 TQFT，它可以区分 r-自旋曲面的所有差分类，我们还证明了与这类 TQFT 相关的弗罗贝尼斯代数必然是非半复数的。

引用次数: 0

Subalgebras of octonion algebras 八元数子代数

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-11 DOI: 10.1016/j.jalgebra.2024.10.004

Norbert Knarr, Markus J. Stroppel

For an arbitrary unitary octonion algebra, we determine all subalgebras. It turns out that every subalgebra of dimension less than four is associative, while every subalgebra of dimension greater than four is not associative. In any split octonion algebra, there are both associative and non-associative subalgebras of dimension four. Except for one-dimensional subalgebras spanned by idempotents, any two isomorphic subalgebras are in the same orbit under automorphisms.

对于任意的单元八元数代数，我们可以确定所有的子代数。结果发现，维数小于四的每个子代数都是联立的，而维数大于四的每个子代数都不是联立的。在任何分裂八元数代数中，都存在维数为四的联立子代数和非联立子代数。除了幂等子所跨的一维子代数外，任何两个同构的子代数在自动变形下都在同一轨道上。

引用次数: 0

Relative monadicity 相对单一性

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-11 DOI: 10.1016/j.jalgebra.2024.08.040

Nathanael Arkor , Dylan McDermott

We establish a relative monadicity theorem for relative monads with dense roots in a virtual equipment, specialising to a relative monadicity theorem for enriched relative monads. In particular, for a dense

-functor

j : A \to E

, a

-functor

r : D \to E

is j-monadic if and only if r admits a left j-relative adjoint and creates j-absolute colimits. This provides a refinement of the classical monadicity theorem – characterising those categories whose objects are given by those of E equipped with algebraic structure – in which the arities of the algebraic operations are valued in A. In particular, when

j = 1

, we recover a formal monadicity theorem. Furthermore, we examine the interaction between the pasting law for relative adjunctions and relative monadicity. As a consequence, we derive necessary and sufficient conditions for the (j-relative) monadicity of the composite of a

-functor with a (j-relatively) monadic

-functor.

我们为虚拟设备中具有稠密根的相对单子建立了一个相对单子性定理，并专门为丰富的相对单子建立了一个相对单子性定理。具体地说，对于一个致密的-矢量 j:A→E, 一个-矢量 r:D→E 是 j-单元的，当且仅当 r 允许一个左 j-相对邻接并产生 j-绝对列。这就提供了经典一元性定理的细化--它描述了那些对象由 E 中配备代数结构的对象给出的范畴--其中代数运算的算术值在 A 中。此外，我们还考察了相对邻接的粘贴定律与相对一元性之间的相互作用。因此，我们推导出了一个-矢量与一个（j-相对）单矢量复合的（j-相对）单矢量的必要条件和充分条件。

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引用次数: 0

Equivalence between invariance conjectures for parabolic Kazhdan-Lusztig polynomials 抛物线卡兹丹-卢兹蒂格多项式不变性猜想之间的等价性

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-11 DOI: 10.1016/j.jalgebra.2024.09.026

Paolo Sentinelli

We prove that the combinatorial invariance conjecture for parabolic Kazhdan-Lusztig polynomials, formulated by Mario Marietti, is equivalent to its restriction to maximal quotients. This equivalence lies at the other extreme in respect to the equivalence, recently proved by Barkley and Gaetz, with the invariance conjecture for Kazhdan-Lusztig polynomials, which turns out to be equivalent to the conjecture for maximal quotients.

我们证明了马里奥-马里埃蒂提出的抛物线卡兹丹-卢兹蒂格多项式的组合不变性猜想等同于其对最大商的限制。这一等价性与巴克利和盖茨最近证明的卡兹丹-卢兹蒂格多项式不变性猜想的等价性处于另一个极端，后者与最大商的猜想等价。

引用次数: 0

A categorification of cluster algebras of type B and C through symmetric quivers 通过对称四元组对 B 型和 C 型簇代数进行分类

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-11 DOI: 10.1016/j.jalgebra.2024.09.015

Azzurra Ciliberti

We express cluster variables of type

B_{n}

and

C_{n}

in terms of cluster variables of type

A_{n}

. Then we associate a cluster tilted bound symmetric quiver Q of type

A_{2 n - 1}

to any seed of a cluster algebra of type

B_{n}

and

C_{n}

. Under this correspondence, cluster variables of type

B_{n}

(resp.

C_{n}

) correspond to orthogonal (resp. symplectic) indecomposable representations of Q. We find a Caldero-Chapoton map in this setting. We also give a categorical interpretation of the cluster expansion formula in the case of acyclic quivers.

我们用 An 类型的簇变量来表达 Bn 和 Cn 类型的簇变量。然后，我们把 A2n-1 类型的簇倾斜约束对称四元组 Q 与 Bn 和 Cn 类型的簇代数的任何种子关联起来。在这种对应关系下，Bn（或 Cn）型簇变量对应于 Q 的正交（或交映）不可分解表示。我们还给出了无环四元组情况下簇扩展公式的分类解释。

引用次数: 0

Pro-C RAAGs Pro-C RAAGs

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-11 DOI: 10.1016/j.jalgebra.2024.09.030

Montserrat Casals-Ruiz , Matteo Pintonello , Pavel Zalesskii

Let

C

be a class of finite groups closed under taking subgroups, quotients, and extensions with abelian kernel. The right-angled Artin pro-

C

group

G_{Γ}

(pro-

C

RAAG for short) is the pro-

C

completion of the right-angled Artin group

G (Γ)

associated with the finite simplicial graph Γ.

In the first part, we describe structural properties of pro-

C

RAAGs. Among others, we describe the centraliser of an element and show that pro-

C

RAAGs satisfy the Tits' alternative, that standard subgroups are isolated, and that 2-generated pro-p subgroups of pro-

C

RAAGs are either free pro-p or free abelian pro-p.

In the second part, we characterise splittings of pro-

C

RAAGs in terms of the defining graph. More precisely, we prove that a pro-

C

RAAG

G_{Γ}

splits as a non-trivial direct product if and only if Γ is a join and it splits over an abelian pro-

C

group if and only if a connected component of Γ is a complete graph or it has a complete disconnecting subgraph. We then use this characterisation to describe an abelian JSJ decomposition of a pro-

C

RAAG, in the sense of Guirardel and Levitt [9].

设 C 是一类在取子群、商和扩展下封闭的有限群，具有无边内核。直角阿尔丁原 C 群 GΓ（简称原 C RAAG）是与有限简单图Γ相关联的直角阿尔丁群 G(Γ)的原 C 完成。在第一部分中，我们描述了 pro-C RAAGs 的结构性质。其中，我们描述了元素的中心化，并证明了 pro-C RAAGs 满足 Tits' 备选，标准子群是孤立的，并且 pro-C RAAGs 的 2 个生成的 pro-p 子群要么是自由 pro-p 要么是自由无边 pro-p.在第二部分中，我们从定义图的角度描述了 pro-C RAAGs 的分裂。更准确地说，我们证明了当且仅当 Γ 是一个连接时，亲 C RAAG GΓ 分裂为一个非三维直积；当且仅当 Γ 的一个连接成分是一个完整图或它有一个完整的断开子图时，它分裂于一个无性亲 C 群。然后，我们根据 Guirardel 和 Levitt [9] 的观点，利用这一特征描述亲 C RAAG 的无边 JSJ 分解。

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引用次数: 0

Galois theory and homology in quasi-abelian functor categories 准阿贝尔函数范畴中的伽罗瓦理论和同源性

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-10-11 DOI: 10.1016/j.jalgebra.2024.09.031

Nadja Egner

Given a finite category

T

, we consider the functor category

A^{T}

, where

A

can be any quasi-abelian category. Examples of quasi-abelian categories are given by any abelian category but also by non-exact additive categories as the categories of torsion(-free) abelian groups, topological abelian groups, locally compact abelian groups, Banach spaces and Fréchet spaces. In this situation, the categories of various internal categorical structures in

A

, such as the categories of internal n-fold groupoids, are equivalent to functor categories

A^{T}

for a suitable category

T

. For a replete full subcategory

S

T

, we define

F

to be the full subcategory of

A^{T}

whose objects are given by the functors

F : T \to A

with

F (T) = 0

for all

T \notin S

. We prove that

F

is a torsion-free Birkhoff subcategory of

A^{T}

. This allows us to study (higher) central extensions from categorical Galois theory in

A^{T}

with respect to

F

and generalized Hopf formulae for homology.

给定一个有限范畴 T，我们考虑函数范畴 AT，其中 A 可以是任何准阿贝尔范畴。准阿贝尔范畴的例子可以是任何无性范畴，也可以是非完全相加范畴，如无扭（-free）无性群、拓扑无性群、局部紧密无性群、巴拿赫空间和弗雷谢特空间等范畴。在这种情况下，A 中各种内部分类结构的范畴，如内部 n 折叠群的范畴，等价于合适范畴 T 的函子范畴 AT。我们将证明 F 是 AT 的无扭 Birkhoff 子类。这样，我们就可以研究 AT 中关于 F 的分类伽罗瓦理论的（高）中心扩展以及同调的广义霍普夫公式。

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引用次数: 0

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全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

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Journal of Algebra

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