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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

二面体群代数的Hopf-Ore扩张

上Yetter-Drinfeld模

朱虹，陈惠香

扬州大学数学科学学院，扬州　220002

摘要：设𝑘是特征为零的代数闭域，𝐷

𝑛

是阶为2𝑛的二面体群，其中𝑛为正的偶数。本文中主要

研究𝑘𝐷

𝑛

的Hopf-Ore扩张𝐴(𝑛, 0)上的Yetter-Drinfeld模。本文描述了𝐴(𝑛, 0)上单Yetter

Drinfeld模的结构和性质，还给出了𝐴(𝑛, 0)上单Yetter-Drinfeld模的同构分类。

关键词：代数，二面体群，Hopf-Ore扩张，Yetter-Drinfeld模

中图分类号： O 153

Yetter-Drinfeld Modules over The Hopf Ore

Extension of The Group Algebra of Dihedral

Group

ZHU Hong , CHEN Hui-Xiang

School of Mathematical Science, Yangzhou University, Yangzhou 225002,

Abstract: Let 𝑘 be an algebraically closed ﬁeld of characteristic zero, and 𝐷

𝑛

be the

dihedral group of order 2𝑛, where 𝑛 is a positive even integer. In this paper, the

Yetter-Drinfeld modules over the Hopf-Ore extension 𝐴(𝑛, 0) of 𝑘𝐷

𝑛

are investigated. The the

structures and properties of simple Yetter-Drinfeld modules over 𝐴(𝑛, 0) are described, and

all simple Yetter-Drinfeld modules over 𝐴(𝑛, 0) are classiﬁed up to isomorphism.

Key words: Algebra, Dihedral group, Hopf-Ore extension, Yetter-Drinfeld module.

0 Introduction and Preliminaries

Yetter-Drinfeld modules over a Hopf algebra 𝐻 were introduced in [1], where they were

called crossed bimodules. When the antipode 𝑆 of 𝐻 is bijective, the category of Yetter-

基金项目： NSF of China (11171291)，Doctorate fundation, Ministry of Education of China (200811170001)

作者简介： Zhu Hong（1983-），female，Doctor，major research direction：algebra. Correspondence author：

Chen Hui-Xiang（1960-），male，professor，major research direction：algebra.

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

Drinfeld modules over 𝐻 is a braided monoidal category. Hence, each Yetter-Drinfeld module

provides a solution to the Yang-Baxter equation (see [2, 3]). If 𝐻 is ﬁnite dimensional, then

a Yetter-Drinfeld module over 𝐻 is the same as a module over the Drinfeld double 𝐷(𝐻). In

recent years a serious eﬀort has beeb made to understand and classify Hopf algebras over an

algebraically closed ﬁeld of characteristic zero. Many papers have deﬁned interesting pointed

Hopf algebras by generators and relations. Beattie, Dascalescu and Grunenfelder described a

general construction in [4] which produces pointed co-Frobenius Hopf algebras starting with

a group algebra of a abelian group by repeated Ore extensions. They gave an isomorphism

theorem which gives necessary and suﬃcient conditions for some Ore extension Hopf algebras

to be isomorphic. However, the isomorphism theorem cannot handle all Ore extension Hopf

algebras. Beattie extended the theorem in [5] to cover all such Ore extension Hopf algebras.

Panov considered in [6] some Ore extension starting with a Hopf algebras to produce new

Hopf algebras. He also gave a classiﬁcation theorem for such Hopf Ore extensions. Wang

considered in [7] a Hopf Ore extension of the group algebra of Dihedral group, and constructed

and classiﬁed all ﬁnite dimensional simple modules over the Hopf Ore extension.

Let 𝑛 be a positive integer with 𝑛 ⩾ 2. Dihedral group 𝐷

𝑛

is a group of order 2𝑛, which is

generated by 𝑎, 𝑏 subject to the relations 𝑎

𝑛

= 𝑏

= 1 and (𝑎𝑏)

= 1. Let 𝑘 be an algebraically

closed ﬁeld of characteristic zero. When 𝑛 = 2𝑑 is even, there is a Hopf Ore extension of 𝑘𝐷

𝑛

denoted by 𝐴(𝑛, 0) (see below for the structure). In this paper, we will construct all simple

Yetter-Drinfeld modules over 𝐴(𝑛, 0), and classify them.

We will organize the paper as follows. In the rest of the section, we recall the structures

of 𝐴(𝑛, 0) and all ﬁnite dimensional simple modules over 𝐴(𝑛, 0). These materials come from

reference [7]. In Section 1, we ﬁrst show that any simple 𝐴(𝑛, 0)-module is ﬁnite dimensional.

Then we recall the deﬁnition of a Yetter-Drinfeld module over a Hopf algebra 𝐻 (simply, YD

𝐻-module), and the YD 𝐻-module structure of 𝐿 ⊗ 𝐻 for a left 𝐻-module. We also discuss

some properties of the YD 𝐴(𝑛, 0)-submodules of 𝐿 ⊗ 𝐻 generated by simple subcomodules,

where 𝐿 is a left simple 𝐴(𝑛, 0)-module. Let 𝐿 be a left simple 𝐴(𝑛, 0)-module. Then a right

simple 𝐴(𝑛, 0)-subcomodule has the form 𝑘(𝑙 ⊗𝑔) for some 𝑙 ∈ 𝐿 and 𝑔 ∈ 𝐷

𝑛

. In Sections 2 and

3, we discuss the structure and properties of the YD 𝐴(𝑛, 0)-module 𝐴(𝑛, 0) ⋅ (𝑙 ⊗ 𝑔) for 𝑑 to be

odd and even respectively. We give a necessary and suﬃcient condition for 𝐴(𝑛, 0) ⋅ (𝑙 ⊗ 𝑔) to

be a simple YD 𝐴(𝑛, 0)-module. We also give a necessary and suﬃcient condition for such two

YD 𝐴(𝑛, 0)-modules to be isomorphic. Then using the classiﬁcation of simple 𝐴(𝑛, 0)-modules,

we classify all simple YD 𝐴(𝑛, 0)-modules.

Throughout, we work over a ﬁxed algebraically closed ﬁeld 𝑘 of characteristic zero. All

Hopf algebras and modules are deﬁned over 𝑘; all maps are 𝑘-linear; dim, ⊗ and Hom stand

for dim

𝑘

, ⊗

𝑘

and Hom

𝑘

, respectively. All modules are left modules, and comodules are right

comodules. For the theory of Hopf algebras, we refer to [3]. Let 𝑘

denote the group 𝑘∖{0}.

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

Let 𝜔 ∈ 𝑘 be a primitive 𝑛-th root of unity, where 𝑛 = 2𝑑 is even.

𝐴(𝑛, 0) is generated, as an associative 𝑘-algebra, by 𝑎, 𝑏, 𝑥, subject to the relations

𝑎

𝑛

= 𝑏

= 1, (𝑎𝑏)

= 1,

𝑥𝑎 = −𝑎𝑥, 𝑥𝑏 = 𝑏𝑥.

It has a Hopf algebra structure determined by

Δ(𝑎) = 𝑎 ⊗ 𝑎, Δ(𝑏) = 𝑏 ⊗ 𝑏,

Δ(𝑥) = 𝑥 ⊗ 1 + 𝑎

𝑑

⊗ 𝑥,

𝜀(𝑎) = 𝜀(𝑏) = 1, 𝜀(𝑥) = 0,

𝑆(𝑎) = 𝑎

−1

, 𝑆(𝑏) = 𝑏

−1

, 𝑆(𝑥) = −𝑎

𝑑

𝑥.

𝐴(𝑛, 0) has a 𝑘-basis {𝑥

𝑠

𝑎

𝑖

𝑏

𝑗

∣ 0 ⩽ 𝑖 ⩽ 𝑛 − 1, 0 ⩽ 𝑗 ⩽ 1, 𝑠 ⩾ 0}, and it is a ℕ-graded

algebra with the grading 𝐴(𝑛, 0) =



𝑠⩾0

𝐴(𝑛, 0)

(𝑠)

, where 𝐴(𝑛, 0)

(𝑠)

is a subspace spanned by

{𝑥

𝑠

𝑎

𝑖

𝑏

𝑗

∣ 0 ⩽ 𝑖 ⩽ 𝑛 − 1, 0 ⩽ 𝑗 ⩽ 1} and 𝐴(𝑛, 0)

(0)

= 𝑘𝐷

𝑛

, the coradical of 𝐴(𝑛, 0).

We know from [7] that 𝐴(𝑛, 0) has ﬁve types of ﬁnite-dimensional simple modules up to

isomorphism. Let ⟨−1⟩ = {1, −1} be the subgroup of the multiplicative group 𝑘

generated by

−1, and 𝑘

/⟨−1⟩ be the quotient group of 𝑘

modulo ⟨−1⟩. These ﬁnite dimensional simple

modules can be described as follows:

1 − dimensional simple modules 𝑘

𝜒

𝑖

, 1 ⩽ 𝑖 ⩽ 4: Let {1

𝑖

} be a basis of 𝑘

𝜒

𝑖

. Then 𝑘

𝜒

𝑖

corresponds to the algebra map 𝜒

𝑖

: 𝐴(𝑛, 0) → 𝑘 given by 𝜒

𝑖

(𝑥) = 0 and

𝜒

(𝑎) = 1, 𝜒

(𝑎) = −1, 𝜒

(𝑎) = −1,

𝜒

(𝑏) = 1, 𝜒

(𝑏) = −1, 𝜒

(𝑏) = 1, 𝜒

(𝑏) = −1.

2 − dimensional simple modules 𝑇

𝑖

, 1 ⩽ 𝑖 ⩽ 𝑑 − 1: 𝑇

𝑖

has a basis {𝑣

𝑖

, 𝑣

𝑖

} such that the

corresponding algebra map 𝜙

𝑖

: 𝐴(𝑛, 0) → 𝑀

(𝑘) is given by

𝜙

𝑖

(𝑎) =



𝜔

𝑖

0 𝜔

−𝑖



, 𝜙

𝑖

(𝑏) =



0 1

1 0



, 𝜙

𝑖

(𝑥) =



0 0



2 − dimensional simple modules 𝑇 (𝜉, ±1), 𝜉 ∈ 𝑘

/⟨−1⟩: 𝑇 (𝜉, ±1) has a basis {𝑣

𝜉,±1

, 𝑣

𝜉,±1

}

such that the corresponding algebra map 𝜙

𝜉,±1

: 𝐴(𝑛, 0) → 𝑀

(𝑘) is given by

𝜙

𝜉,±1

(𝑎) =



1 0

0 −1



, 𝜙

𝜉,±1

(𝑏) = ±



1 0

0 1



, 𝜙

𝜉,±1

(𝑥) = 𝜉



0 1

1 0



2 − dimensional simple modules 𝑇 (𝜉), 𝜉 ∈ 𝑘

, here 𝑑 is even: 𝑇 (𝜉) has a basis {𝑣

𝜉

, 𝑣

𝜉

}

such that the corresponding algebra map 𝜙

𝜉

: 𝐴(𝑛, 0) → 𝑀

(𝑘) is given by

𝜙

𝜉

(𝑎) =



𝜔

𝑑/2

0 𝜔

−𝑑/2



, 𝜙

𝜉

(𝑏) =



0 1

1 0



, 𝜙

𝜉

(𝑥) = 𝜉



0 1

1 0



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