[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()

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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

Author(s):

Ma Tianshui¹;  2;  Wang Shuanhong¹

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

Keywords:

oriented quantum coalgebra;  twisted oriented quantum coalgebra;  knot invariant

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.

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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