arXiv math/04071652004-07math.RTmath.RA

Representations and Cocycle Twists of Color Lie Algebras

Xiao-Wu Chen, Sergei D. Silvestrov, Fred van Oystaeyen

备注：: 18 pages, with an concrete example
发表信息：: Algebras and Representation Theory, Vol.9, No.6 (2006), 633-650

英文摘要：

We study relations between finite-dimensional representations of color Lie algebras and their cocycle twists. Main tools are the universal enveloping algebras and their FCR-properties (finite-dimensional representations are completely reducible.) Cocycle twist preserves the FCR-property. As an application, we compute all finite dimensional representations (up to isomorphism) of the color Lie algebra $sl_2^c$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Graded-simple algebras and cocycle twisted loop algebras

The loop algebra construction by Allison, Berman, Faulkner, and Pianzola, describes graded-central-simple algebras with split centroid in terms of central simple algebras graded by a quotient of the original grading group. Here the restriction on the centroid is removed, at the expense of allowing some deformations (cocycle twists) of the loop algebras.

2018-10-26math.RAmath.RT

Cocycle Twists and Extensions of Braided Doubles

It is well known that central extensions of a group G correspond to 2-cocycles on G. Cocycles can be used to construct extensions of G-graded algebras via a version of the Drinfeld twist introduced by Majid. We show how 2-cocycles can be defined for an abstract monoidal category C, following Panaite, Staic and Van Oystaeyen. A braiding on C leads to analogues of Nichols algebras in C, and we explain how the recent work on twists of Nichols algebras by Andruskiewitsch, Fantino, Garcia and Vendramin fits in this context. Furthermore, we propose an approach to twisting the multiplication in braid…

2013-02-10math.QAmath.RT

Cocycle twisting of E(n)-module algebras and applications to the Brauer group

We classify the orbits of coquasi-triangular structures for the Hopf algebra E(n) under the action of lazy cocycles and the Hopf automorphism group. This is applied to detect subgroups of the Brauer group $BQ(k,E(n))$ of E(n) that are isomorphic. For a triangular structure $R$ on E(n) we prove that the subgroup $BM(k,E(n),R)$ of $BQ(k,E(n))$ arising from $R$ is isomorphic to a direct product of $BW(k)$, the Brauer-Wall group of the ground field $k$, and $Sym_n(k)$, the group of $n \times n$ symmetric matrices under addition. For a general quasi-triangular structure $R'$ on E(n) we construct a …

2004-04-01math.RAmath.RT

Yetter-Drinfeld modules under cocycle twists

We give an explicit formula for the correspondence between simple Yetter-Drinfeld modules for certain finite-dimensional pointed Hopf algebras $H$ and those for cocycle twists $H^σ$ of $H$. This implies an equivalence between modules for their Drinfeld doubles. To illustrate our results, we consider the restricted two-parameter quantum groups ${\mathfrak{u}}_{r,s}({\mathfrak{sl}}_n)$ under conditions on the parameters guaranteeing that ${\mathfrak{u}}_{r,s}({\mathfrak{sl}}_n)$ is a Drinfeld double of its Borel subalgebra. We determine explicit correspondences between ${\mathfrak{u}}_{r,s}({\ma…

2009-10-27math.QAmath.RT

Filiform Z2xZ2-color Lie superalgebras

We continue the study of the filiform Z2xZ2-color Lie superalgebras. All of them can be obtained by using infinitesimal deformations, i.e. cocycles. In this work we give the total dimension of such cocycles (for any dimensions n, m, p and t of the Z2xZ2-color Lie superalgebras). Also, we give a basis of such cocycles in some generic and concrete cases.

2014-07-21math.RT

Cocycle deformations and Galois objects of semisimple Hopf algebras of dimension $16$

In this article, we determine cocycle deformations and Galois objects of non-commutative and non-cocommutative semisimple Hopf algebras of dimension $16$. We show that these Hopf algebras are pairwise twist inequivalent mainly by calculating their higher Frobenius-Schur indicators, and that except three Hopf algebras which are cocycle deformations of dual group algebras, none of them admit non-trivial cocycle deformations.

2019-06-12math.RT

更多相关论文收起更多论文

Non-existence of Hopf orders for a twist of the alternating and symmetric groups

We prove the non-existence of Hopf orders over number rings for two families of complex semisimple Hopf algebras. They are constructed as Drinfel'd twists of group algebras for the following groups: $A_n$, the alternating group on $n$ elements, with $n \geq 5$; and $S_{2m}$, the symmetric group on $2m$ elements, with $m \geq 4$ even. The twist for $A_n$ arises from a $2$-cocycle on the Klein four-group contained in $A_4$. The twist for $S_{2m}$ arises from a $2$-cocycle on a subgroup generated by certain transpositions which is isomorphic to $\mathbb{Z}_2^m$. This provides more examples of com…

2018-11-19math.QAmath.RAmath.RT

The nonvanishing first Hochschild cohomology of twisted finite simple group algebras

Let $G$ be a finite simple group and $k$ be an algebraically closed field of prime characteristic dividing the order of $G$. We show that for all $2$-cocycles $α\in Z^2(G;k^\times)$, the first Hochschild cohomology group of the twisted group algebra $HH^1(k_αG)$ is nonzero.

2022-07-08math.GRmath.KTmath.RA

非交换领域中扭曲的导论：聚焦于代数、表示论与几何

A Primer on Twists in the Noncommutative Realm Focusing on Algebra, Representation Theory, and Geometry

本文系统介绍了非交换代数中通过自同构、双代数2-上循环及张量积等方式扭曲乘法结构的几类技术，并侧重通过丰富实例阐明其分类与性质。

2023-09-05math.QAmath.RAmath.RT

复反射群与有理Cherednik代数的表示的扭

Twists of representations of complex reflection groups and rational Cherednik algebras

该研究证明代数分解与诱导表示的上循环扭的一般性结果，将其应用于反射群与有理Cherednik代数，刻画相关模的扭并证明非交换代数的Chevalley定理类似结论。

2025-01-12math.QAmath.RT

Non-existence of integral Hopf orders for twists of several simple groups of Lie type

Let $p$ be a prime number and $q=p^m$, with $m \geq 1$ if $p \neq 2,3$ and $m>1$ otherwise. Let $Ω$ be any non-trivial twist for the complex group algebra of $\mathbf{PSL}_2(q)$ arising from a $2$-cocycle on an abelian subgroup of $\mathbf{PSL}_2(q)$. We show that the twisted Hopf algebra $(\mathbb{C} \mathbf{PSL}_2(q))_Ω$ does not admit a Hopf order over any number ring. The same conclusion is proved for the Suzuki groups, and for $\mathbf{SL}_3(p)$ when the twist stems from an abelian $p$-subgroup. This supplies new families of complex semisimple (and simple) Hopf algebras that do not admit …

2022-08-02math.GRmath.QAmath.RA