Journal of Algebra最新文献

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Infinite-dimensional Lie bialgebras via affinization of perm bialgebras and pre-Lie bialgebras 通过 perm 双桥和前列双桥的肤射化实现无穷维列双桥

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-09-24 DOI: 10.1016/j.jalgebra.2024.09.006

Yuanchang Lin , Peng Zhou , Chengming Bai

It is known that the operads of perm algebras and pre-Lie algebras are the Koszul dual each other and hence there is a Lie algebra structure on the tensor product of a perm algebra and a pre-Lie algebra. Conversely, we construct a special perm algebra structure and a special pre-Lie algebra structure on the vector space of Laurent polynomials such that the tensor product with a pre-Lie algebra and a perm algebra being a Lie algebra structure characterizes the pre-Lie algebra and the perm algebra respectively. This is called the affinization of a pre-Lie algebra and a perm algebra respectively. Furthermore we extend such correspondences to the context of bialgebras, that is, there is a bialgebra structure for a perm algebra or a pre-Lie algebra which could be characterized by the fact that its affinization by a quadratic pre-Lie algebra or a quadratic perm algebra respectively gives an infinite-dimensional Lie bialgebra. In the case of perm algebras, the corresponding bialgebra structure is called a perm bialgebra, which can be independently characterized by a Manin triple of perm algebras as well as a matched pair of perm algebras. The notion of the perm Yang-Baxter equation is introduced, whose symmetric solutions give rise to perm bialgebras. There is a correspondence between symmetric solutions of the perm Yang-Baxter equation in perm algebras and certain skew-symmetric solutions of the classical Yang-Baxter equation in the infinite-dimensional Lie algebras induced from the perm algebras. In the case of pre-Lie algebras, the corresponding bialgebra structure is a pre-Lie bialgebra which is well-constructed. The similar correspondences for the related structures are given.

众所周知，永恒代数和前李代数的操作数互为科斯祖尔对偶，因此永恒代数和前李代数的张量积上存在一个李代数结构。反过来，我们在劳伦多项式的向量空间上构造了一个特殊的前列代数结构和一个特殊的前列代数结构，使得前列代数和前列代数的张量积分别表征前列代数和前列代数的列代数结构。这分别被称为前列代数和后列代数的肤射化。此外，我们还将这种对应关系扩展到双代数的范畴，即存在一种双代数结构，它可以表征为：一个前列代数或一个后列代数分别与一个二次前列代数或一个二次后列代数缀合后得到一个无穷维的李双代数。在永恒代数的情况下，相应的双代数结构称为永恒双代数，它可以由永恒代数的马宁三元组以及永恒代数的匹配对独立表征。引入了永恒杨-巴克斯特方程的概念，其对称解产生了永恒双代数。烫发杨-巴克斯特方程在烫发代数中的对称解与经典杨-巴克斯特方程在烫发代数诱导的无穷维李代数中的某些偏对称解之间存在对应关系。在前李代数的情况下，相应的双代数结构是一个结构良好的前李双代数。本文给出了相关结构的类似对应关系。

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引用次数: 0

Homogeneous quandles with abelian inner automorphism groups 具有非等边内自变群的均质曲

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-09-17 DOI: 10.1016/j.jalgebra.2024.09.004

Takuya Saito , Sakumi Sugawara

In this paper, we give a characterization of homogeneous quandles with abelian inner automorphism groups. In particular, we show that such a quandle is expressed as an abelian extension of a trivial quandle. Our construction is a generalization of the recent work by Furuki and Tamaru, which gives a construction of disconnected flat quandles.

在本文中，我们给出了具有非等边内自变群的均质 quandle 的特征。特别是，我们证明了这样的 quandle 可以表示为微不足道的 quandle 的无边扩展。我们的构造是对 Furuki 和 Tamaru 的最新研究成果的推广，后者给出了断开的平面 quandle 的构造。

引用次数: 0

On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties for GLn 论 GLn 的 p-adic Deligne-Lusztig varieties 的模ℓ 同调

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-09-17 DOI: 10.1016/j.jalgebra.2024.08.033

Jakub Löwit

In 1976, Deligne and Lusztig realized the representation theory of finite groups of Lie type inside étale cohomology of certain algebraic varieties. Recently, a p-adic version of this theory started to emerge: there are p-adic Deligne–Lusztig spaces, whose cohomology encodes representation theoretic information for p-adic groups – for instance, it partially realizes the local Langlands correspondence with characteristic zero coefficients. However, the parallel case of coefficients of positive characteristic

ℓ \neq p

has not been inspected so far. The purpose of this article is to initiate such an inspection. In particular, we relate cohomology of certain p-adic Deligne–Lusztig spaces to Vignéras's modular local Langlands correspondence for

{GL}_{n}

1976 年，德莱尼和卢兹蒂格在某些代数变种的 étale 同调内实现了有限列群的表示理论。最近，这一理论的 p-adic 版本开始出现：存在 p-adic Deligne-Lusztig 空间，其同调包含 p-adic 群的表示理论信息--例如，它部分实现了特征零系数的局部朗兰兹对应关系。然而，正特征 ℓ≠p 的系数的平行情况迄今为止还没有被研究过。本文的目的就是启动这样的研究。特别是，我们将某些 p-adic Deligne-Lusztig 空间的同调与 GLn 的 Vignéras 模块局部朗兰兹对应关系联系起来。

引用次数: 0

Tilting theory for finite dimensional 1-Iwanaga-Gorenstein algebras 有限维 1-岩永-哥伦布代数的倾斜理论

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-09-17 DOI: 10.1016/j.jalgebra.2024.08.034

Yuta Kimura , Hiroyuki Minamoto , Kota Yamaura

We study tilting objects of the stable category

{\underline{CM}}^{Z} A

of graded Cohen-Macaulay modules over a finite dimensional graded Iwanaga-Gorenstein algebra A. We first show that if there exists a tilting object in

{\underline{CM}}^{Z} A

, then its endomorphism algebra always has finite global dimension. Next, to study the existence of a tilting object, we introduce a numerical invariant

g (A)

. In the case where A is 1-Iwanaga-Gorenstein, we give a sufficient condition on

g (A)

for the existence of a tilting object. As an application, for a truncated preprojective algebra

Π {(Q)}_{w}

of a tree quiver Q, we prove that

{\underline{CM}}^{Z} Π {(Q)}_{w}

always admits a tilting object.

我们研究有限维分级岩永-戈伦斯坦代数 A 上的分级科恩-麦考莱模块稳定范畴 CM_ZA 的倾斜对象。我们首先证明，如果 CM_ZA 中存在一个倾斜对象，那么它的内构代数总是具有有限全维。接下来，为了研究倾斜对象的存在，我们引入了数值不变式 g(A)。在 A 是 1-Iwanaga-Gorenstein 的情况下，我们给出了 g(A) 存在倾斜对象的充分条件。作为应用，对于树状四元组 Q 的截断前投影代数Π(Q)w，我们证明 CM_ZΠ(Q)w 总是承认一个倾斜对象。

引用次数: 0

Ore localisation for differential graded rings; towards Goldie's theorem for differential graded algebras 微分级数环的矿石局部化；迈向微分级数代数的戈尔迪定理

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-09-17 DOI: 10.1016/j.jalgebra.2024.08.032

Alexander Zimmermann

We study Ore localisation of differential graded algebras. Further we define dg-prime rings, dg-semiprime rings, and study the dg-nil radical of dg-rings. Then, we define dg-essential submodules, dg-uniform dimension, and apply all this to a dg-version of Goldie's theorem on prime dg-rings.

我们研究了微分级数代数的 Ore 局部化。此外，我们还定义了dg-原环、dg-半原环，并研究了dg-环的dg-无根。然后，我们定义了dg-本质子模、dg-统一维数，并将所有这些应用于素数dg环上戈尔迪定理的dg-版本。

引用次数: 0

Metric ultraproducts of groups — Simplicity, perfectness and torsion 群的公设超积 - 简单性、完备性和扭转性

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-09-17 DOI: 10.1016/j.jalgebra.2024.09.005

Jakub Gismatullin , Krzysztof Majcher , Martin Ziegler

We characterise the simplicity of metric ultraproducts of a family of metric groups. We also present several new examples of simple groups, such as metric ultraproducts of finite and infinite symmetric groups, linear groups, and interval exchange transformation groups. Using similar methods, we also examine concepts such as genericity, perfectness, and torsion.

我们描述了一个度量群族的度量超积的简单性。我们还介绍了几个新的简单群实例，如有限和无限对称群、线性群和区间交换变换群的度量超积。我们还使用类似的方法研究了泛函、完备性和扭转等概念。

引用次数: 0

On the polynomiality conjecture of cluster realization of quantum groups 论量子群簇实现的多项式猜想

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-09-17 DOI: 10.1016/j.jalgebra.2024.08.031

Ivan Chi-Ho Ip , Jeff York Ye

In this paper, we give a sufficient and necessary condition for a regular element of a quantum cluster algebra

O_{q} (X)

to be universally polynomial. This resolves several conjectures by the first author on the polynomiality of the cluster realization of quantum group generators in different families of positive representations.

在本文中，我们给出了量子簇代数 Oq(X) 的正则元普遍多项式的充分必要条件。这就解决了第一作者关于不同正表示族中量子群生成器的簇实现的多项式性的几个猜想。

引用次数: 0

Seminormal forms for the Temperley-Lieb algebra Temperley-Lieb 代数的半正态形式

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-09-10 DOI: 10.1016/j.jalgebra.2024.09.003

Katherine Ormeño Bastías , Steen Ryom-Hansen

Let ${TL}_{n}^{Q}$ be the rational Temperley-Lieb algebra, with loop parameter 2. In the first part of the paper we study the seminormal idempotents $E_{t}$ for ${TL}_{n}^{Q}$ for $t$ running over two-column standard tableaux. Our main result is here a concrete combinatorial construction of $E_{t}$ using Jones-Wenzl idempotents ${JW}_{k}$ for ${TL}_{k}^{Q}$ where $k \leq n$ .

In the second part of the paper we consider the Temperley-Lieb algebra ${TL}_{n}^{F_{p}}$ over the finite field $F_{p}$ , where $p > 2$ . The KLR-approach to ${TL}_{n}^{F_{p}}$ gives rise to an action of a symmetric group $S_{m}$ on ${TL}_{n}^{F_{p}}$ , for some $m < n$ . We show that the $E_{t}$ 's from the first part of the paper are simultaneous eigenvectors for the associated Jucys-Murphy elements for $S_{m}$ . This leads to a KLR-interpretation of the p-Jones-Wenzl idempotent ${}^{p}{JW}_{n}$ for ${TL}_{n}^{F_{p}}$ , that was introduced recently by Burull, Libedinsky and Sentinelli.

设 TLnQ 是有理滕伯里-李布代数，循环参数为 2。在本文的第一部分，我们研究了 TLnQ 中运行于两列标准表格的 t 的半正态幂级数 Et。在论文的第二部分，我们考虑了有限域 Fp 上的 Temperley-Lieb 代数 TLnFp，其中 p>2。对于某个 m<n，TLnFp 的 KLR 方法产生了对称群 Sm 对 TLnFp 的作用。我们证明，本文第一部分的 Et 是 Sm 的相关 Jucys-Murphy 元素的同时特征向量。这引出了 Burull、Libedinsky 和 Sentinelli 最近提出的 TLnFp 的 p-Jones-Wenzl 瞬态 JWnp 的 KLR 解释。

引用次数: 0

Characteristic subgroups and the R∞-property for virtual braid groups 虚拟辫状群的特征子群和 R∞ 属性

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-09-06 DOI: 10.1016/j.jalgebra.2024.09.002

Karel Dekimpe , Daciberg Lima Gonçalves , Oscar Ocampo

<div>Let <math><mi>n</mi><mo>≥</mo><mn>2</mn></math>. Let <math><mi>V</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math> (resp. <math><mi>V</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math>) denote the virtual braid group (resp. virtual pure braid group), let <math><mi>W</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math> (resp. <math><mi>W</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math>) denote the welded braid group (resp. welded pure braid group) and let <math><mi>U</mi><mi>V</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math> (resp. <math><mi>U</mi><mi>V</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math>) denote the unrestricted virtual braid group (resp. unrestricted virtual pure braid group). In the first part of this paper we prove that, for <math><mi>n</mi><mo>≥</mo><mn>4</mn></math>, the group <math><mi>V</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math> and for <math><mi>n</mi><mo>≥</mo><mn>3</mn></math> the groups <math><mi>W</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math> and <math><mi>U</mi><mi>V</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math> are characteristic subgroups of <math><mi>V</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math>, <math><mi>W</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math> and <math><mi>U</mi><mi>V</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math>, respectively. In the second part of the paper we show that, for <math><mi>n</mi><mo>≥</mo><mn>2</mn></math>, the virtual braid group <math><mi>V</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math>, the unrestricted virtual pure braid group <math><mi>U</mi><mi>V</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math>, and the unrestricted virtual braid group <math><mi>U</mi><mi>V</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math> have the R∞-property. As a consequence of the technique used for few strings we also prove that, for <math><mi>n</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn></math>, the welded braid group <math><mi>W</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math> has the R∞-property and that for <math><mi>n</mi><mo>=</mo><mn>2</mn></math> the corresponding pure braid groups have the R∞-property. On the other hand for <math><mi>n</mi><mo>≥</mo><mn>3</mn></math> it is unknown if the R∞-property holds or not for the virtual pure braid group <math><mi>V</mi><msub><mrow><m

设 n≥2。让 VBn（或 VPn）表示虚辫群（或虚纯辫群），让 WBn（或 WPn）表示焊接辫群（或焊接纯辫群），让 UVBn（或 UVPn）表示无限制虚辫群（或无限制虚纯辫群）。在本文的第一部分，我们将证明，对于 n≥4，群 VPn，对于 n≥3，群 WPn 和 UVPn 分别是 VBn、WBn 和 UVBn 的特征子群。在论文的第二部分，我们证明了在 n≥2 时，虚辫群 VBn、无限制虚纯辫群 UVPn 和无限制虚辫群 UVBn 具有 R∞ 属性。作为对少数弦使用的技术的结果，我们还证明，对于 n=2、3、4，焊接辫状群 WBn 具有 R∞属性，对于 n=2，相应的纯辫状群具有 R∞属性。另一方面，当 n≥3 时，虚拟纯辫状群 VPn 和焊接纯辫状群 WPn 的 R∞ 属性是否成立还是未知数。

引用次数: 0

Central extensions of axial algebras 轴代数的中心扩展

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-09-05 DOI: 10.1016/j.jalgebra.2024.09.001

Ivan Kaygorodov , Cándido Martín González , Pilar Páez-Guillán

In this article, we develop a further adaptation of the method of Skjelbred-Sund to construct central extensions of axial algebras. We use our method to prove that all axial central extensions (with respect to a maximal set of axes) of complex simple finite-dimensional Jordan algebras are split, and that all non-split axial central extensions of dimension $n \leq 4$ over an algebraically closed field of characteristic not 2 are Jordan. Also, we give a classification of 2-dimensional axial algebras and describe some important properties of these algebras.

在本文中，我们进一步发展了斯基尔布雷德-桑德（Skjelbred-Sund）的方法，以构建轴向代数的中心扩展。我们用我们的方法证明了复简单有限维乔丹布拉的所有轴中心扩展（关于轴的最大集）都是分裂的，并且证明了在特征非 2 的代数闭域上维数 n≤4 的所有非分裂轴中心扩展都是乔丹的。此外，我们还给出了二维轴代数的分类，并描述了这些代数的一些重要性质。

引用次数: 0

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