Zhao Xiaofan Wang Shuanhong
(Department of Mathematics, Southeast University, Nanjing 211189, China)
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A note on ribbon elements of Hopf group-coalgebras
Zhao Xiaofan Wang Shuanhong
(Department of Mathematics, Southeast University, Nanjing 211189, China)
LetGbe a discrete group with a neutral element andHbe a quasitriangular HopfG-coalgebra over a fieldk. Then the relationship betweenG-grouplike elements and ribbon elements ofHis considered. First, a list of useful properties of a quasitriangular HopfG-coalgebra and its Drinfeld elements are proved. Secondly, motivated by the relationship between the grouplike and ribbon elements of a quasitriangular Hopf algebra, a special kind ofG-grouplike elements ofHis defined. Finally, using the Drinfeld elements, a one-to-one correspondence between the specialG-grouplike elements defined above and ribbon elements is obtained.
quasitriangular HopfG-coalgebra;G-grouplike element; ribbon element; Drinfeld element
In the theory of the classical Hopf algebras[1-2], one of the celebrated results is the theory of ribbon Hopf algebras, which plays an important role in constructing invariants of framed links embedded in 3-dimensional space[3]. One important aspect of ribbon Hopf algebras is the relationship between grouplike elements and ribbon elements[4].
As a generalization of ordinary Hopf algebras, Hopf group-coalgebras related to homotopy quantum field theories were introduced by Turaev in Ref.[5]. A purely algebraic study of Hopf group-coalgebras, such as the main properties of quasitriangular and ribbon Hopf group-coalgebras, can be found in Refs.[6-9].
In this paper, we consider the following question: for a groupG, how to use a special kind ofG-grouplike elements to describe the ribbon elements of a HopfG-coalgebra.
Throughout this paper, we letGbe a discrete group with a neutral element 1 andkbe a field. Assume thatHis a HopfG-coalgebra overk. Denote the set of allG-grouplike elements ofHbyG(H).
Definition 1AHopfG-coalgebraH=({Hα},Δ,ε,S) is said to be crossed provided it is endowed with a familyφ={φβ:Hα→Hβαβ-1}α,β∈Gofk-linear maps (the crossing) such that for allα,β,γ∈G, 1)φβis an algebra isomorphism; 2) (φβ?φβ)Δα,γ=Δβαβ-1,βγβ-1φβ; 3)εφβ=ε; 4)φαβ=φαφβ.
Definition 2 A quasitriangular HopfG-coalgebra is a crossed HopfG-coalgebraH=({Hα},Δ,ε,S,φ) endowed with a familyR={Rα,β∈Hα?Hβ}α,β∈Gof invertible elements (theR-matrix) such that for allα,β,γ∈G, andx∈Hαβ,
Rα,βΔα,β(x)=σβ,α(φα-1?idHα)Δαβα-1,α(x)Rα,β
(idHα?Δβ,γ)(Rα,βγ)=(Rα,γ)1β3(Rα,β)12γ(Δα,β?idHγ)(Rαβ,γ)=[(idHα?φβ-1)(Rα,βγβ-1)]1β3(Rβ,γ)α23
(φβ?φβ)(Rα,γ)=Rβαβ-1,βγβ-1
Remark 1 LetH=({Hα,mα,1α},Δ,ε,S,φ,R) be a quasitriangular HopfG-coalgebra. The generalized Drinfeld elements ofHare defined byμα=mα(Sα-1φα?idHα)σα,α-1(Rα,α-1)∈Hα, for anyα∈G.
(2)
Proof We first check the identity (1). For anyα,β∈G
Next we show the proof of the identity (2). For anyα,β∈G, we have the following computations:
Lemma 3 LetH=({Hα},Δ,ε,S,φ,R) be a quasitriangular HopfG-coalgebra. Then for anyα,β∈G,
Proof Using Lemma 1 and Lemma 2, for anyα,β∈G, we compute
This completes the proof of the lemma.
Theorem 1 Suppose thatH=({Hα},Δ,ε,S,φ,R) is a quasitriangular HopfG-coalgebra. Then there is a one-to-one correspondence betweenEandFdefined as above.
Let us prove the third condition. We compute
HencePis well defined. Secondly, we show thatPhas an inverse map. Define a mapQ:F→EbyQ(θ)=μθ={μαθα∈Hα}α∈G, for anyθ∈F. Clearly,PQ=idF,QP=idE. Following Ref.[6], we know thatQis well defined. This completes the proof of the theorem.
[1]Sweedler M.Hopfalgebras[M]. New York: Benjamin, 1969.
[2]Montgomery S.Hopfalgebrasandtheiractionsonrings[M]. Rhode Island: American Mathematical Society, 1993.
[3]Reshetikhin N Y, Turaev V G. Ribbon graphs and their invariants derived from quantum groups [J].CommMathPhys, 1990, 127(1): 1-26.
[4]Kauffman L H, Radford D E. A necessary and sufficient condition for a finite-dimensional Drinfel’d double to be a ribbon Hopf algebra [J].JAlgebra, 1993, 159(1): 98-114.
[5]Turaev V G. Homotopy field theory in dimension 3 and crossed group-categories[EB/OL]. (2000)[2013-07-01].http://arxiv.org/abs/math/0005291.
[6]Virelizier A. Hopf group-coalgebras [J].JPureApplAlgebra, 2002, 171(1): 75-122.
[7]Virelizier A. Graded quantum groups and quasitriangular Hopf group-coalgebras [J].CommAlgebra, 2004, 33(9): 3029-3050.
[8]Wang S H. Group entwining structures and group coalgebras Galois extensions [J].CommAlgebra, 2004, 32(9): 3417-3436.
[9]Wang S H. Group twisted smash products and Doi-Hopf modules for T-coalgebras [J].CommAlgebra, 2004, 32(9): 3437-3458.
關(guān)于Hopf群余代數(shù)ribbon元的注記
趙曉凡 王栓宏
(東南大學(xué)數(shù)學(xué)系, 南京 211189)
設(shè)G是一個(gè)帶有單位元的離散群,H是域k上的擬三角HopfG-余代數(shù). 考慮了H的G-群像元和ribbon元之間的關(guān)系. 首先證明了擬三角HopfG-余代數(shù)以及它的Drinfeld元的一些重要性質(zhì). 受到Hopf代數(shù)中群像元和ribbon元之間關(guān)系的啟發(fā), 定義了一類特殊的G-群像元. 最后利用Drinfeld元得到了所定義的特殊的G-群像元和ribbon元之間的一個(gè)一一對(duì)應(yīng)關(guān)系.
擬三角HopfG-余代數(shù);G-群像元; ribbon元; Drinfeld元
O153.3
Foundation items:The National Natural Science Foundation of China (No.11371088), the Natural Science Foundation of Jiangsu Province (No.BK2012736), the Fundamental Research Funds for the Central Universities (No.KYZZ0060).
:Zhao Xiaofan, Wang Shuanhong.A note on ribbon elements of Hopf group-coalgebras[J].Journal of Southeast University (English Edition),2015,31(2):294-296.
10.3969/j.issn.1003-7985.2015.02.024
10.3969/j.issn.1003-7985.2015.02.024
Received 2013-10-27.
Biographies:Zhao Xiaofan (1986—), female, graduate; Wang shuanhong (corresponding author), male, doctor, professor, shuanhwang@seu.edu.cn.
Journal of Southeast University(English Edition)2015年2期