[1] Ma Tianshui, Wang Shuanhong,. Oriented quantum coalgebra structure on the tensor productof an oriented quantum coalgebra with itself [J]. Journal of Southeast University (English Edition), 2009, 25 (2): 286-288. [doi:10.3969/j.issn.1003-7985.2009.02.030]

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Oriented quantum coalgebra structure on the tensor productof an oriented quantum coalgebra with itself()

定向量子余代数与其自身张量积上的定向量子余代数结构

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

25

Issue:

2009 2

Page:

286-288

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2009-06-30

Info

Title:

Oriented quantum coalgebra structure on the tensor productof an oriented quantum coalgebra with itself

定向量子余代数与其自身张量积上的定向量子余代数结构

Author(s):

Ma Tianshui^{1, 2}, Wang Shuanhong¹

¹ Department of Mathematics, Southeast University, Nanjing 211189, China
² Department of Mathematics, Henan Normal University, Xinxiang 453007, China

马天水^{1, 2}, 王栓宏¹

¹东南大学数学系, 南京 211189; ²河南师范大学数学系, 新乡 453007

Keywords:

oriented quantum coalgebra;  twisted oriented quantum coalgebra;  knot invariant

定向量子余代数;  扭曲定向量子余代数;  扭结不变量

PACS:

O153.3

DOI:

10.3969/j.issn.1003-7985.2009.02.030

Abstract:

Oriented quantum algebras(coalgebras)are generalizations of quasitriangular Hopf algebras(coquasitriangular Hopf algebras)and account for regular isotopy invariants of oriented 1-1 tangles, oriented knots and links.Let(H, σ, D, U)be an oriented quantum coalgebra over the field k.Then(H⊗H, φ, D⊗D, U⊗U)is a trivial oriented quantum coalgebra structure on the tensor product of coalgebra H with itself, where φ(a⊗b, c⊗d)=σ(a, c)σ(b, d).This paper presents the oriented quantum coalgebra structure(H⊗H, (~overσ), D⊗D, U⊗U)on coalgebra H⊗H, where (~overσ)(a⊗b, c⊗d)=σ^-11(d₁1, a₁1)σ(a₂2, c₁1)σ^-11(d₂2, b₁1)σ(b₂2, c₂2).So a nontrivial oriented quantum coalgebra structure is obtained and it is dual to Radford’s results in the paper “On the tensor product of an oriented quantum algebra with itself” published in 2007.Theoretically, the results of this paper are important in constructing the invariants of oriented knots and links.

定向量子代数(定向量子余代数)是拟三角Hopf代数(余拟三角Hopf代数)的推广并且可以得到定向1-1缠绕、定向扭结和连接的正则合痕不变量.令(H, σ, D, U)为域k上的定向量子余代数, 则(H⊗H, φ, D⊗D, U⊗U)为余代数H与其自身张量积上一个平凡的定向量子余代数结构, 其中φ(a⊗b, c⊗d)=φ(a, c)φ(b, d).本文给出余代数H⊗H上的一种定向量子余代数结构(H⊗H, (~overσ), D⊗D, U⊗U), 其中(~overσ)(a⊗b, c⊗d)=σ^-11(d₁1, a₁1)·σ(a₂2, c₁1)σ^-11(d₂2, b₁1)σ(b₂2, c₂2).进一步得到定向量子余代数与其自身张量积上的一种非平凡的定向量子余代数结构, 这一结果对偶于Radford发表于2007年的《On the tensor product of an oriented quantum algebra with itself》一文中的结论.理论上本文的结果对于构造定向扭结和连接不变量是非常重要的.

References:

[1] Drinfeld V G.Quantum groups[C]//Proceedings of the International Congress of Mathematicians.Berkeley, CA, USA, 1987:798-820.
[2] Radford D E.On the tensor product of an oriented quantum algebra with itself[J].J Knot Theory Ramifications, 2007, 16(7):929-957.
[3] Kauffman L H, Radford D E.Oriented quantum algebras, categories and invariants of knots and links[J].J Knot Theory Ramifications, 2001, 10(7):1047-1084.
[4] Kauffman L H, Radford D E.Invariants of 3-manifolds derived from finite dimensional Hopf algebras[J].J Knot Theory Ramifications, 1995, 4(1):131-162.
[5] Kauffman L H, Radford D E.Oriented quantum algebras and invariants of knots and links[J].J Algebra, 2001, 246(1):253-291.
[6] Kauffman L H, Radford D E.Quantum algebras, quantum coalgebras, invariants of 1-1 tangles and knots[J].Comm Algebra, 2000, 28(11):5101-5156.
[7] Montgomery S.Hopf algebras and their actions on rings [M].Providence, RI:American Mathematical Society, 1993.

Memo

Memo:

Biographies: Ma Tianshui(1977—), male, graduate;Wang Shuanhong(corresponding author), male, doctor, professor, shuanhwang2002@yahoo.com.
Foundation item: The National Natural Science Foundation of China(No.10871042).
Citation: Ma Tianshui, Wang Shuanhong. Oriented quantum coalgebra structure on the tensor product of an oriented quantum coalgebra with itself[J].Journal of Southeast University(English Edition), 2009, 25(2):286-288.

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