本文的目的是描述Zagliki闭代数的结构，它为有限域上可表示代数的研究提供了一个有用的推广.我们的研究结果包括一个版本的韦德伯恩的主要定理，以及更明确的描述使用表示，在“胶合”韦德伯恩组件。最后，我们构造了“通有”的Zagliki闭代数，它的描述比有限维代数的通有代数的描述要复杂得多。特别注意有限域上的无限维代数。

2011-09-22math.RA

如何计算有限维结合代数的Wedderburn分解

How to compute the Wedderburn decomposition of a finite-dimensional associative algebra

这是一个调查文件的算法，已在过去25年中开发的显式计算的结构的有限维的结合代数在有限域或代数数域。这种建设性的方法是由Friedl和Ronyai于1985年提出的，后来由Cohen、de Graaf、Eberly、Giesbrecht、Ivanyos、Kuronya和Wales发展起来。用n = 2的n元部分变换半群PT_n的有理半群代数来说明这些算法，推广了全变换半群和对称逆半群，推广了对称群S_n。

2010-08-11math-phmath.MPmath.RA

Finite semisimple group algebra of a normally monomial group

In this paper, the complete algebraic structure of finite semisimple group algebra of a normally monomial group is described. The main result is illustrated by computing the explicit Wedderburn decomposition of finite semisimple group algebras of various normally monomial groups. The automorphism groups of these group algebras are also determined.

2017-07-26math.RA

The structure of Hopf algebras giving Hopf-Galois structures on Quaternionic extensions

Let $L/F$ be a Galois extension of fields with Galois group isomorphic to the quaternion group of order $ 8 $. We describe all of the Hopf-Galois structures admitted by $ L/F $, and determine which of the Hopf algebras that appear are isomorphic as Hopf algebras. In the case that $ F $ has characteristic zero we also determine which of these Hopf algebras are isomorphic as $ F $-algebras and explicitly compute their Wedderburn-Artin decompositions.

2018-12-05math.RA

中心单代数的构造性基础理论

Constructive basic theory of central simple algebras

本文针对中心单代数理论中Wedderburn定理在构造上无效的问题，通过证明其动态版本，为Skolem-Noether定理等基础结果提供了构造性证明，并给出了Becher定理的初等构造性证明。

2023-06-30math.LOmath.RA

Radical and weight of skew braces and their applications to structure groups of solutions of the Yang-Baxter equation

We define the radical and weight of a skew left brace and provide some basic properties of these notions. In particular, we obtain a Wedderburn type decomposition for Artinian skew left braces. Furthermore, we prove analogues of a theorem of Wiegold, a theorem of Schur and its converse in the context of skew left braces. Finally, we apply these results to detect torsion in the structure group of a finite bijective non-degenerate set-theoretic solution of the Yang-Baxter equation.

2020-01-29math.GRmath.RA

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Structure of group rings and the group of units of integral group rings: an invitation

During the past three decades fundamental progress has been made on constructing large torsion-free subgroups (i.e. subgroups of finite index) of the unit group $\U (\Z G)$ of the integral group ring $\Z G$ of a finite group $G$. These constructions rely on explicit constructions of units in $\Z G$ and proofs of main results make use of the description of the Wedderburn components of the rational group algebra $\Q G$. The latter relies on explicit constructions of primitive central idempotents and the rational representations of $G$. It turns out that the existence of reduced two degree repres…

2020-08-26math.RA

Polyadization of algebraic structures

A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented in block diagonal matrix form (resulting in the Wedderburn decomposition), a general form of polyadic structures is given by block-shift matrices. We combine these forms to get a general shape of semisimple nonderived polyadic structures ("double" decomposition of two kinds). We then introduce the polyadization concept (a "polyadic constructor") according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary stru…

2022-09-08hep-thmath.GRmath.RA

Infinite-dimensional diagonalization and semisimplicity

We characterize the diagonalizable subalgebras of End(V), the full ring of linear operators on a vector space V over a field, in a manner that directly generalizes the classical theory of diagonalizable algebras of operators on a finite-dimensional vector space. Our characterizations are formulated in terms of a natural topology (the "finite topology") on End(V), which reduces to the discrete topology in the case where V is finite-dimensional. We further investigate when two subalgebras of operators can and cannot be simultaneously diagonalized, as well as the closure of the set of diagonaliza…

2016-10-21math.RA

环上Chevalley群的泛局部化结构

Structure of Chevalley groups over rings via universal localization

本文研究交换环$R$上Chevalley群$G(R)$的结构.推广和改进了下列结果：（1）标准、相对和多相对交换子公式;（2）[相对] $K_1$的幂零结构;（3）交换子的有界字长.为此，我们扩大了初等群，构造了一个通用元素的扩展初等群，并在通用环中使用本地化。关键的一步是构造对应于主理想的主同余子群的类元。

2015-11-23math.RA