数学物理学报

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弱斜配对双代数和弱相关Long双代数

张良云   

  1. 南京农业大学数学系 南京 210095; 南京大学数学系 南京 210008
  • 收稿日期:2004-12-30 修回日期:2006-03-10 出版日期:2006-08-25 发布日期:2006-08-25
  • 通讯作者: 张良云
  • 基金资助:
    国家自然科学基金(10571153)、中国科学博士后基金(2005037713)江苏省高等学校自然科学

Weak Skew Paired Bialgebras and Weak Relative Long Bialgebras

Zhang Liangyun   

  1. College of Science, Nanjing Agricultural University, Nanjing 210095
  • Received:2004-12-30 Revised:2006-03-10 Online:2006-08-25 Published:2006-08-25
  • Contact: Zhang Liangyun

摘要:

该文在弱双代数H上给出了扭曲积(Hσ,σ)成为弱双代数的充分必要条件.设[B,H,τ]是一个弱斜配对, 并且τ可逆,则在某个条件下弱双交叉积BτH是一个弱双代数. 如果(B,H,σ)是弱相关Long双代数, 并且σ可逆,则弱双交叉积BOPσH可以被构造. 它的乘法是:
(xh)(yg)=Σσ(y1,h1)y2xh2gσ1(y3,h3), 特别地, 如果(B,H,σ)是相关Long双代数, 则(B^{OP \bowtie_\sigma H,\beta)是Long双代数当且仅当对任意b,dBOP;g,H,
Σσ1(b,g2)σ(d,g1)=Σσ1(b,g1)σ(d,g2), 其中BH的子Hopf代数,β定义为β(bσhcσg)=εH(h)εBOP(c)σ1(b,g). 对于Sweedler 4维Hopf代数H, 作者给出一个例子说明:
此弱双交叉积(BOPσH,β)不仅是一个Long双代数,
而且是一个非可换和非余可换的8维Hopf代数. 最后, 设B,H都是弱双代数, σ:BHk是一个线性映射, 作者给出了(B,σ,,ΔB)是弱相关右(H,B) -重模代数的充分必要条件.

关键词: 弱Doi双代数, 弱双交叉积, 弱相关Long双代数, 弱相关重模代数

Abstract: This paper gives a sufficient and
necessary condition for given twisted product
(Hσ,σ) to be a weak bialgebra. If [B,H,τ] are weak skew paired bialgebras and τ is invertible,
then, under some condition, the weak
bicrossed product BτH is a weak bialgebra. If (B,H,σ) is a weak relative Long bialgebra and σ
invertible, then the weak bicrossed product BOPσH can be constructed. Espically, for the Sweedler 4-dimensional
Hopf algebra H4, the author gives an example to show that
(BOPσH4,β) is not only a Long bialgebra
but also a non-commutative and non-cocommutative 8-dimensional
Hopf algebra, where B is a sub-Hopf algebra of H4. If B and
H are weak bialgebras, and σ:BHk is
a linear map, then a sufficient and necessary condition
for (B,σ,,ΔB) to be a weak right
relative (H,B)-dimodule algebra is given.

Key words: Weak skew paired bialgebra, Weak bicrossed product, Weak relative long bialgebra, Weak relative dimodule algebra

中图分类号: 

  • 16W30