Advances in Pure Mathematics, 2017, 7, 366-374 http://www.scirp.org/journal/apm ISSN Online: 2160-0384 ISSN Print: 2160-0368

From Braided Infinitesimal Bialgebras to Braided Lie Bialgebras

Shengxiang Wang1,2

1Department of Mathematics, Nanjing University, Nanjing, China 2School of Mathematics and Finance, Chuzhou University, Chuzhou, China

How to cite this paper: Wang, S.X. (2017) Abstract From Braided Infinitesimal Bialgebras to Braided Lie Bialgebras. Advances in Pure The present paper is a continuation of [1], where we considered braided infi- Mathematics, 7, 366-374. nitesimal Hopf algebras (i.e., infinitesimal Hopf algebras in the Yetter-Drin- https://doi.org/10.4236/apm.2017.77023 H  feld category H for any Hopf algebra H), and constructed their Drinfeld Received: May 22, 2017 double as a generalization of Aguiar’s result. In this paper we mainly investi- Accepted: July 14, 2017 gate the necessary and sufficient condition for a braided infinitesimal bialge- Published: July 17, 2017 H bra to be a braided Lie bialgebra (i.e., a Lie bialgebra in the category H  ). Copyright © 2017 by author and Scientific Research Publishing Inc. Keywords This work is licensed under the Creative Commons Attribution International Braided Infinitesimal Bialgebra, Braided Lie Bialgebra, Yetter-Drinfeld License (CC BY 4.0). Category, Balanceator http://creativecommons.org/licenses/by/4.0/ Open Access

1. Introduction

An infinitesimal bialgebra is a triple ( Am,,∆) , where ( Am, ) is an associative algebra (possibly without unit), ( A,∆) is a coassociative coalgebra (possibly without counit) such that

∆( xy) = xy1 ⊗ y 21 +⊗ x xyxyA 2, , ∈ .

Infinitesimal bialgebras were introduced by Joni and Rota in [2], called infini- tesimal coalgebra there, in the context of the calculus of divided differences [3]. In combinatorics, they were further studied in [4] [5] [6]. Aguiar established the basic theory of infinitesimal bialgebras in [7] [8] by investigating several examples and the notions of antipode, Drinfeld double and the associative Yang- Baxter equation keeping close to ordinary Hopf algebras. In [9], Yau introduced the notion of Hom-infinitesimal bialgebras and extended Aguiar’s main results in [7] [8] to Hom-infinitesimal bialgebras. One of the motivations of studying infinitesimal bialgebras is that they are

DOI: 10.4236/apm.2017.77023 July 17, 2017 S. X. Wang

closely related to Drinfeld’s Lie bialgebras (see [10]). The cobracket ∆ in a Lie bialgebra is a 1-cocycle in Chevalley-Eilenberg cohomology, which is a 1-cocycle in Hochschild cohomology (i.e., a derivation) in a infinitesimal bialgebra. So the compatible condition in a infinitesimal bialgebra can be seen as an associative analog of the cocycle condition in a Lie bialgebra. Motivated by [1], in which we considered infinitesimal Hopf algebras in the Yetter-Drinfeld categories, called braided infinitesimal Hopf algebras, the natural idea is whether we can obtain braided Lie bialgebras (called generalized H-Lie bialgebras in [11] [12]) from braided infinitesimal Hopf algebras. This becomes our motivation of writing this paper. To give a positive answer to the question above, we organize this paper as follows. In Section 1, we recall some basic definitions about Yetter-Drinfeld modules and braided Lie bialgerbas. In Section 2, we introduce the notion of the balanceator of a braided infinitesimal bialgerba and show that a braided infinitesimal bialgerba gives rise to a braided Lie bialgerba if and only if the balanceator is symmetric (see Theorem 2.3).

2. Preliminaries

In this paper, k always denotes a fixed field, often omitted from the notation. We use Sweedler’s ([13]) notation for the comultiplication: ∆=⊗(hhh) 12, for all hH∈ . Let H be a Hopf algebra. We denote the category of left H-modules by H H  . Similarly, we have the category  of left H-comodules. For a left H- comodules (M , ρ ) , we also use Sweedler’s notation: ρ (mm) =(−1) ⊗ m0 , for all mM∈ . A left-left Yetter-Drinfeld module M is both a left H-module and a left H- comodule satisfying the compatibility condition ⊗⋅ = ⋅ ⊗ ⋅ hm1(−1) hm 20( hm 1)(−10) h 2( hm 1) (2.1) for all hH∈ and mM∈ . The equation (1.1) is equivalent to

ρ (h⋅= m) hm1(−1) S( h 3) ⊗⋅( h 20 m ). (2.2)

H By [14] [15], the left-left Yetter-Drinfeld category H  is a braided monoi- dal category whose objects are Yetter-Drinfeld modules, morphisms are both left

H-linear and H-colinear maps, and its braiding C−−, is given by

CMN, ( mn⊗) = m(−10) ⋅⊗ nm( ) ,

H H for all mM∈∈H and nN∈∈H . H Let A be an object in H , the braiding τ is called symmetric on A if the following condition holds:

⋅ ⋅⊗ ⋅=⊗ (ab(−−11) ) a0 ( ab( ) ) ab, (2.3) (−10) which is equivalent to the following condition: ⋅⊗ = ⊗−1 ⋅ a(−−11) ba00 b S( b( ) ) a, (2.4)

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for any ab,.∈ A H In the category H , we call an (co)algebra simply if it is both a left H- module (co)algebra and a left H-comodule (co)algebra. For more details about (co)module-(co)algebras, the reader can refer to [16] [17]. H A braided Lie algebra ([11]) in H , called generalized H-Lie algebra there, H is an object L in H together with a bracket operation [,:] LL⊗→ L, which H is a morphism in H satisfying ′=−⋅∈ ′′ (1) H-anti-commutativity: [ll, ]  l(−1) l, l0 , ll, L , (2) H-Jacobi identity: {ll⊗⊗′ l ′′} +{(τ ⊗1)( 1 ⊗ τ)(ll ⊗⊗′ l ′′ )} +{(1 ⊗ττ)( ⊗ 1)(ll ⊗⊗′ l ′′ )} =0,

for all ll,,′ l ′′ ∈ L, where {ll⊗⊗′ l ′′} denotes lll,,[ ′ ′′ ] and τ the braiding for L. H Let A be an associative algebra in H . Assume that the braiding is symmetric on A. Define =−⋅ ∈ [ab, ] ab( a(−1) b) a0 , ab, A .

Then ( A,,[ ]) is a braided Lie algebra (see [11]). H A braided Lie coalgebra ([12]) Γ is an object in H together with a linear H map δ : Γ→Γ⊗Γ (called the cobracket), which is also a morphism in H subject to the following conditions: (1) H-anti-cocommutativity: δ= − τδ , (2) H-coJacobi identity: (id+( id ⊗τ)( τ ⊗ id) +⊗( τ id)( id ⊗ τ))( id ⊗ δδ) =0,

where τ denotes the braiding for L. H Dually, let (C,∆) be a coassociative coalgebra in H . Assume that the braiding on C is symmetric. Define δ :,C→⊗ CC by

c c1⊗− c 2 c11(− ) ⋅⊗ c2 ccC 10 , ∈.

H Then (C,δ ) is a braided Lie coalgebras in H (see [12]). H A braided Lie bialgebra ([18]) is (L,,,[ ] δ ) in H , where (L,,[ ]) is a braided Lie algebra, and (L,δ ) is a braided Lie coalgebra, such that the compatibility condition holds: δ[x,, y] =(([ ] ⊗ id)( id ⊗+ δ) ( id ⊗[,])( τδ ⊗ id)( id ⊗))( id ⊗− id τ)( x ⊗ y), x, y ∈ L ,

where τ denotes the braiding for L.

3. Main Results

In this section, we will study the relation between braided infinitesimal bialge- bras and braided Lie bialgebras as a generalization of Aguiar’s result in [8]. H Let ( Am,,∆) be a braided ε-bialgebra in H . For any xyz,,∈ A, define an action of A on AA⊗ by → ⊗ = ⊗− ⋅⊗ ⋅ x( yz) xyzx(−−1) y( x 01( ) zx) 00.

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H Then the action → is a morphism in H . In fact, for any xyz,,∈ A and hH∈ , we have

hx1⋅→ hyz 2 ⋅( ⊗) = hx 1 ⋅→( hyhz 23 ⋅⊗ ⋅) =hxhy ⋅ ⋅ ⊗ hz ⋅ − hx ⋅ ⋅ hy ⋅⊗ hx ⋅ ⋅ hzhx ⋅ ⋅ ( 12)( ) ( 3) ( 1)(−−1) 2(( 1)0( 1) 31)( )00 =hxhy ⋅ ⋅ ⊗ hzhxSh ⋅ − ⋅ hy ⋅⊗ hx ⋅ ⋅ hzhx ⋅ ⋅ ( 1)( 2 ) ( 3) 11 (−1) ( 13) 2 (( 12 0)(−10) 3)( 12 0 ) =hxhy ⋅ ⋅ ⊗ hzhxy ⋅ − ⋅⊗ hx ⋅ ⋅ hzhx ⋅ ⋅ ( 1)( 2) ( 3) 1(−1) (( 20)(−10) 3)( 20)

= ⋅ ⋅ ⊗ ⋅ − ⋅⊗ ⋅ ⋅⋅ (h1 x)( h 2 y) ( h3 z) hx 1 (−−1) y( h21 x01( ) S( h23) h 3 zh)( 22 x 00 ) = ⋅ ⋅ ⊗ ⋅ − ⋅⊗ ⋅ ⋅ (hxhy1)( 2 ) ( hz3) hx 1 (−−1) y( hx2 01( ) zhx)( 3 00 ) = ⋅ ⊗ ⋅ − ⋅⊗ ⋅ ⋅ hxy1 ( ) ( hzhx2) 1 (−−1) yh2 (( x01( ) zx) 00 ) =⋅ ⊗− ⋅⊗ ⋅ h( xy z x(−−1) y( x 01( ) z) x00 ). So → is left H-linear. To show the left H-colinearity of the action →, we compute ρρ→ ⊗ = ⊗− ⋅⊗ ⋅ ( x( yz)) ( xyzx(−−1) y( x 01( ) zx) 00 )

=xyzxyzxy(−1) ( −− 1) ( 1) ⊗0 0 ⊗− 0 (−1) ⋅ x01(−) ⋅ zx 001(−) ⊗ xy( − 1) ⋅ ⊗ x01(−) ⋅ zx000 ( )(−−11) ( )( ) ( ) 0( ) 0

= ⊗ ⊗− ⊗ ⋅⊗ ⋅ x(−−−1) y( 1) z( 1) xy00 z 0 x(−−−−−−−11) y( 1) Sx( ( 13) ) x( 1) 4 z( 1) Sx( ( 1) 6) x( 1) 7 x( − 1) 2 y0 ( x( −1) 5 z0) x 0 = ⊗ ⊗− ⊗ ⋅⊗ ⋅ xyz(−1) ( −− 11) ( ) xyzxyz00 0 (−1111) ( −−) ( ) x( − 12) y0 ( x(− 13) zx0) 0, and ⊗→ρ ⊗ ⊗ = ⊗→ ⊗ ⊗ ⊗ (id) ( x y z) ( id)( x(−1) y( −− 11) z( ) x0 y 00 z ) = ⊗ ⊗− ⊗ ⋅⊗ ⋅ xyz(−1) ( −− 1) ( 1) xyzxyz0 0 0 (−1) ( −− 1) ( 1) x 01( −) y0 ( x001(−) zx0) 000 = ⊗ ⊗− ⊗ ⋅⊗ ⋅ xyz(−1) ( −− 11) ( ) xyzxyz00 0 (−1111) ( −−) ( ) x( − 12) y0 ( x(− 13) zx0) 0, as desired. Definition 2.1. Let ( Am,,∆) be a braided infinitesimal bialgebra and τ the braiding of A. The map BAAAA: ⊗→⊗ defined by Bxy( , ) = x →∆τ( y) + ττ( y →∆( x)), xy, ∈ A , (3.1) is called the balanceator of A. The balanceator B is called symmetric if BB= τ . The braided infinitesimal bialgebra A is called balanced if B ≡ 0 on A. Condition (2.1) can be written as follows: = ⋅ ⊗ − ⋅⊗ ⋅ Bxyxyyyxyyxyx( , ) ( 11(−) 2) 10 (−−1) 11( ) 2 ( 01(−) 10) 00

+⋅⋅⊗⋅ − ⋅⋅ ⋅⊗ ( xy(−1) ) x01 ( xyx( −−− 111) ) 02 ( xy( ) ) xxy01 ( ( ) ) x02 (−−1) 0( 10) Obviously, ⋅ = ⋅ ⋅ ⊗ − ⋅ ⊗ + ⋅⊗ Bxyxxyxx( (−1) , 0 ) ( ( −−1) )( 01( ) 02) x 010 ( xyxyxyxy(−1) 1) 0 2 (−1) 1 0 2

− ⋅ ⋅⊗ ⋅ ⋅ ⋅ ( xyx(−−1) ) 01( 1) x02 ( xy( −1) ) x010 ( xy( −1) ) . (−−1) 0( 1) 00

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Lemma 2.2. Let ( Am,,∆) be a braided infinitesimal bialgebra and xy, ∈ A. Assume that the braiding τ on A is symmetric. Then the following equations hold:  ⋅ ⋅⊗ ⋅ ⋅ ⋅=⊗ (1) ( xy(−1) ) x01 ( xy( −− 11) ) xxy02 ( ( ) ) xxy1 2 ,  (−−1)  0( 1)  00

⋅ ⋅ ⋅⊗ = ⋅ ⊗ (2) ( xyx(−1) ) 011( −−) xxy02 ( ( 1) ) x010 x11( −) ( xyx2) 10 , (−10)

(3) x(−11) ⋅ y1 ⋅ x(−12) ⋅ yx20 ⊗ x(−11) ⋅= y1 x(−−1) y 1( 1) ⋅ yx20 ⊗ y 10. ( )(−10) (( ) ) ( ) ( ) Proof. (1) Since the braiding τ on A is symmetric, for all xy, ∈ A, we have

xy(−−11) ⋅ ⋅⊗ x0 xy( ) ⋅ =⊗ xy, then ( )(−10) ( )

⊗ ∆⊗ ⋅ ⋅ ⊗ ⋅ = ⊗ ∆⊗ ⊗ (id m)( id)( x(−−11) y) x0 ( x( ) y) ( id m)( id)( x y) (−10)

that is,

⋅ ⋅⊗ ⋅ ⋅ ⋅ =⊗ ( xy(−1) ) x01 ( xy( −− 11) ) xxy02 ( ( ) ) xxy1 2 . (−−1) 0( 1) 00

So (1) holds. (2) To show the Equation (2.2), we need the following computation:

⋅ ⋅ ⋅⊗ ( xyx(−1) ) 01( −− 1) xxy02 ( ( 1) ) x010 (−10) = ⋅ ⋅ ⋅⊗ ( xx11(−−) 21( ) y) x 101(−) x20 ( xx 11( −−) 21( ) y) x100 (−10) = ⋅ ⋅⊗ ( x111211(−) x( −−) y( 1) Sx( 213( −) ) Sx( 1213( −) ) x 101( −) x20 )( x 112212( −) x( −) y0) x 100

= ⋅ ⋅⊗ ( x111211(−) x( −−) y( 1) Sx( 213( −) ) x20 )( x 112212(−) x( −) y0) x 10

= ⋅⋅ ⋅⊗ x111(−−) ( x 21( ) y) xx20 11(−)2( xy 21(− ) ) x10 (−1) ( 0 )  = ⋅ ⋅ ⋅ ⋅ ⊗= ⋅ ⊗ x11(−−) ( xyxxy 21( ) ) 20 ( 21(−) ) xxxyx10 11(−) ( 2) 10 . (−10)

The last equality holds since τ is symmetric on A. Hence (2) holds. (3) Finally, we check the Equation (2.3) as follows:

xy(−−11) ⋅1 xyxxy( 12) ⋅20 ⊗⋅( −11) 1 ( )(−10) (( ) ) ( ) = ⋅ ⋅ ⊗⋅ ( x(−−1) 11 y 1( 1) Sx( ( − 1) 13 )) (( x( −1) 2 y2) x 0 ) x(−1) 12 y10 = ⋅ ⋅ ⊗⋅ ( xySxxyxxy(−−11) 1( 1) ( ( − 13) )) (( ( − 14) 2) 0 ) (−12) 10 = ⋅ ⋅⊗ ⋅ ( x(−−1) 11 y 1( 1) 1 Sx( ( − 1) 32) x( − 1) 4 y2 )( x( −−1) 12 y 1( 1) 2 Sx( ( − 1) 31 ) x0 ) x(−1) 2 y10 = ⋅ ⋅⊗ ⋅ ( x(−−111) y 1( 11) y2 )( x( −−112) y 1( 12) Sx( ( − 13) ) x0 ) x( −12) y10

= ⋅ ⋅ ⋅⊗ ⋅ ( xy(−−11) 1( 1) y2 )( xy(−−12) 10) x 0 ( xy( 12) 10 ) (−10) = ⋅ ⋅ ⋅⊗ ⋅ = ⋅ ⊗ ( xyyxy(−−1) 11( ) 2 )( 01( −) 10) x 00 ( xy01(−) 10 ) ( xyyxy(−−1) 11( ) 2) 0 10 . (−10)

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The last equality holds since τ is symmetric on A. Hence (3) holds as required.  Therem 2.3. Let ( Am,,∆) be a braided infinitesimal bialgebra. Assume that the braiding τ on A is symmetric. Then ( A,,[ ] = mm −τδ , =∆− τ ∆) is a braided Lie bialgebra if and only if BB= τ . Proof. Since ( Am, ) is an associative algebra and ( A,∆) is a coassociative H coalgebra in H , ( A,,[ ] = mm − τ ) is a braided Lie algebra and (,A δτ=∆− ∆ ) is a braided Lie coalgebra. Therefore it remains to check the compatible condition: δ[x,, y] =(([ ] ⊗ id)( id ⊗+ δ) ( id ⊗[,])( τδ ⊗ id)( id ⊗))( id ⊗− id τ)( x ⊗ y), for all xy, ∈ A. In fact, on the one hand, we have

δδ= −⋅ [x, y] ( xy( x(−1) y) x0 ) =−∆ττ −−∆ ⋅ (11) ( xy) ( ) (( x(−1) y) x0 )

=−(1 τ )( x12 ⊗ x y +⊗ xy 12 y ) −−τ ⋅ ⊗ ⋅ + ⋅ ⊗ (1 )(( x(−−11) y1 ) ( x( 12) y2) x 0 ( x( −1) yx) 01 x 02 ) =⊗ + ⊗− ⋅ ⊗ − ⋅⊗ xxyxy1 2 1 yx 2 11(− ) ( xy2) x 10( xy 1)(−10) y 2( xy 1 ) − ⋅ ⊗− ⋅⊗ ⋅ ( x(−1) yx) 01 x 02 ( x( −−11) y1 ) ( x( 12) y2) x 0

+⋅xy(−−11) 1 ⋅ xyxxy( 12) ⋅20 ⊗⋅( −11) 1 ( )(−10) (( ) ) ( ) +⋅ ⋅⊗⋅ (( x(−1) yx) 01 ) x02 (( x(−1) yx) 01 ) . (−1) 0

On the other hand, we have (([,,]⊗id)( id ⊗+δ) ( id ⊗[ ])( τδ ⊗ id)( id ⊗))( id ⊗− id τ)( x ⊗ y)

=(([,,] ⊗id)( id ⊗+δ) ( id ⊗[ ])( τδ ⊗ id)( id ⊗))( xy −( x(− 1) ⋅ y) x 0 ) = ⊗− ⋅ ⊗− ⋅ ⊗ xyyxyxyxyy1 2 ( (−−1) 1) 0 2 ( 11( ) 2) y 10 + ⋅ ⊗− ⋅ ⊗ ( x(−−1) y 11( ) y2) x 0 y 10 ( x(−1) yx) 01 x 02

+ ⋅ ⋅ ⋅ ⊗+ ⋅ ⋅ ⊗ ( xy(−1) ) xxy01 ( ( −1) ) x02 ( xyxx( −−1) )( 01( 1) 02) x 010 (−10) + ⋅⊗ − ⋅ ⋅ ⋅⊗ xyxyxyx(−1) 1 02 ( ( −−1) ) 01( 1) xxy02 ( (−1) ) x010 (−1) 0 − ⋅⊗ ⋅ − ⋅⊗ x(−1) y1 ( x01( −) y2) x 00 x(−−1) y 11( ) y2 xy 0 10

+xyy(−−1) 11( ) ⋅⊗2 x01(−) ⋅ yxxy10 00 −(−1) ⋅ ⋅ x01 ⊗ xyx(−1) ⋅ 02 ( ) ( )(−10) ( )

+ ⋅ ⋅⊗ ⋅ ⋅ ⋅ ( xy(−1) ) x01 ( xy( −− 11) ) xxy02 ( ( ) ) (−−1) 0( 1) 00

+xyx(−1) ⋅01( −− 1) ⋅⊗ x02 xyx( 1) ⋅ 010 ( )(−10) ( )

− ⋅ ⋅⊗ ⋅ ⋅  ⋅ ( xyx(−−1) ) 01( 1) x02 ( xy( −1) ) x010 ( xy(−1) ) . (−−1)  01( )  00

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According to Lemma 2.2, we have ⊗+ ⋅ ⊗ − ⋅ ⊗ xyy1 2 ( xy(−−1) 11( ) yxy2) 0 10 ( x(−1) yx) 01 x 02

− ⋅ ⋅ ⋅⊗ ( xyx(−1) ) 01( −− 1) xxy02 ( ( 1) ) x010 (−10) − ⋅⊗ ⋅ − ⋅⊗ x(−1) y1 ( x01( −) y2) x 00 x(−−1) y 11( ) y2 xy 0 10

+ ⋅ ⋅⊗ ⋅ ⋅ ⋅ ( xy(−1) ) x01 ( xy( −− 11) ) xxy02 ( ( ) ) (−−1) 0( 1) 00 + ⋅ ⋅⊗ ⋅ ( xyx(−1) ) 01( −− 1) x02 ( xyx( 1) ) 010 (−10)

=xy1 ⊗+ y 2 x(−−11) ⋅ y1 ⋅ x( 12) ⋅ y20 x ⊗ x(−11) ⋅ y1 ( )(−1) (( ) ) ( )0 − ⋅ ⊗− ⋅ ⊗ ( x(−−1) y) x01 x 02 x11( ) ( xy2) x 10 − ⋅⊗ ⋅ − ⋅⊗ x(−−11) y1( x( 12) y20) x( xy 1)(−10) y 2( xy 1) +⊗ + ⋅ ⋅ ⊗ ⋅ x1 xy 2 (( x(−−11) yx) 01) x 02 (( x( ) yx) 01 ) (−10) = δ [xy,.]

Therefore, (([,,]⊗id)( id ⊗+δ) ( id ⊗[ ])( τδ ⊗ id)( id ⊗))( id ⊗− id τ)( x ⊗ y)

=δ − ⋅⊗+ ⋅ ⋅ ⋅ ⊗ [xyxyy, ] ( 1(− 1) 2) y 10 ( xy(−−11) ) xxy01 ( ( ) ) x02 (−10)

+xyy(−−1) 11( ) ⋅⊗2 x01(−) ⋅ yxxy10 00 −(−1) ⋅ ⋅ x01 ⊗ xyx(−1) ⋅ 02 ( ) ( )(−10) ( ) + ⋅ ⋅ ⊗ − ⋅ ⊗ + ⋅⊗ ( xyxx(−−1) )( 01( 1) 02) x 010 ( xyxyxyxy(−1) 1) 0 2 (−1) 1 0 2

− ⋅ ⋅⊗ ⋅ ⋅  ⋅ ( xyx(−−1) ) 01( 1) x02 ( xy( −1) ) x010 ( xy(−1) ) (−−1) 01( ) 00 =−δ +⋅ [xy,,] Bxy( ) Bx( (−1) yx ,0 ) =−+δτ[xy,] Bxy( ,) B ( xy ,,)

as desired. We complete the proof.  Corollary 2.4. Let ( Am,,∆) be a braided infinitesimal bialgebra. Assume that the braiding τ on A is symmetric and the balanceator B = 0 . Then ( A,,[ ] = mm −τδ , =∆− τ ∆) is a braided Lie bialgebra. Proof. Straightforward from Theorem 2.3.  Example 2.5. Let q be an 2th root of unit of k and G the cyclic group of order 2 generated by g, H= kG be the group algebra in the usual way. We consider 4 the algebra A4 = kx[ ] ( x) . By [8], A4 is a infinitesimal bialgebra equipped with the comultiplication: ∆(1) = 0, ∆( x) = xx ⊗2 −⊗1 x 3222332 , ∆( x) = x ⊗ x, ∆( x) = x ⊗ x.

Define the left-H-module action and the left-H-comodule coaction of A by gi⋅= x j qx ij j, ρ ( x j) = g j ⊗ x j , i =0,1, j =0,1,2,3.

It is not hard to check that the multiplication and the comultiplicaition are

372 S. X. Wang

both H-linear and H-colinear, therefore A4 is a braided infinitesimal bialgebra. 22 2 2 3 3 Since B( x,2 x) = x ⊗ x − qx ⊗ x − qx ⊗− x qx ⊗−⊗ x x x and τ ⊗= ⋅ =⋅ =⊗ = τ ( xx) ( x(−1) xx) 0 ( gxxqxx) , it is clear that B( xx,,) B( xx) if and only if q =1. If q =1, it is not hard to check that the balanceator is symmetric on A4 . By Theorem 2.3, ( A4 ,,[ ] = mm −τδ , =∆− τ ∆) is a braided Lie bialgebra. Example 2.6. Let q be a 4th root of unit of k. Consider the Hopf algebra H= kG , where G is a cyclic group of order 4 generated by g. Recall from [1] H that AMk= 2 ( ) is a braided infinitesimal bialgebra in H equipped with the comultiplication: ab 0 a  01  01 cd  ∆  =⊗−⊗     cd 0 c  00  00  00  and the H-module action, the H-comodule coaction:

k 22kij( ++) (ij) g⋅= Eij q Eij, ρ( E ij ) = g ⊗ Eij , k = 0,1, 2, 3, ij , = 1, 2.

Since

BE( 11,2 E 21 ) =( E12 ⊗−⊗ E 22 E 11 E 12 ),

BE⋅ EE, = BEE( ,2) =( E ⊗−⊗ E E E), ( 11(−1) 21 110 ) 21 11 22 12 11 11 we claim that the balanceator is not symmetric. By Theorem 2.3,

(Mk2 ( ),,[ ] = mm −τδ , =∆− τ ∆) is not a braided Lie bialgebra, where m is the multiplication of A. ab Let A12=|, ab ∈⊂ k M( k) . It is clear that A1 is both H-stable and 0 a

H-costable, hence A1 is also a braided infinitesimal bialgebra contained in A.

One can check easily that the balanceator B = 0 on A1 . By Corollary 2.4,

( A1,,[ ] = mm −τδ , =∆− τ ∆) is a braided Lie bialgebra.

Acknowledgements

The paper is partially supported by the China Postdoctoral Science Foundation (No. 2015M571725), the Key University Science Research Project of Anhui Province (Nos. KJ2015A294 and KJ2016A545), the outstanding top-notch talent cultivation project of Anhui Province (No. gxfx2017123) and the NSF of Chuzhou University (No. 2015qd01).

References [1] Wang, S.X. and Wang, S.H. (2014) Drinfeld Double for Braided Infinitesimal Hopf Algebras. Communications in Algebra, 42, 2196-2213. https://doi.org/10.1080/00927872.2013.766796 [2] Joni, S.A. and Rota, G.C. (1979) Coalgebras and Bialgebras in Combinatorics. Studies in Applied Mathematics, 61, 93-139. https://doi.org/10.1002/sapm197961293 [3] Hirschhorn, P.S. and Raphael, L.A. (1992) Coalgebraic Foundations of the Method of Divided Differences. Advances in Mathematics, 91, 75-135.

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[4] Aguiar, M. (2002) Infinitesimal Hopf Algebras and the Cd-Index of Polytopes. Dis- crete & Computational Geometry, 27, 3-28. https://doi.org/10.1007/s00454-001-0049-4 [5] Ehrenborg, R. and Readdy, M. (1998) Coproducts and the Cd-Index. Journal of Al- gebraic Combinatorics, 8, 273-299. https://doi.org/10.1023/A:1008614816374 [6] Foissy, L. (2009) The Infinitesimal Hopf Algebra and the Poset of Planar Forests. Journal of Algebraic Combinatorics, 30, 277-309. https://doi.org/10.1007/s10801-008-0163-y [7] Aguiar, M. (2000) Infinitesimal Hopf Algebras. Contemporary Mathematics, 267, 1-30. https://doi.org/10.1090/conm/267/04262 [8] Aguiar, M. (2001) On the Associative Analog of Lie Bialgebras. Journal of Algebra, 244, 492-532. https://doi.org/10.1006/jabr.2001.8877 [9] Yau, D. (2010) Infinitesimal Hom-Bialgebras and Hom-Lie Bialgebras. arXiv. 1001.5000v1. [10] Drinfeld, V.G. (1986) Quantum Groups. Proceedings International Congress of Mathematicians, Berkeley, 798-820. [11] Wang, S.H. (2002) On the Generalized H-Lie Structure of Associative Algebras in Yetter-Drinfeld Categories. Communications in Algebra, 30, 307-325. https://doi.org/10.1081/AGB-120006492 [12] Wang, S.H. and Chen, H.X. (2002) Hopf-Galois Coextensions and Braided Lie Coalgebras. Progress in Natural Science, 12, 264-270. [13] Sweedle, M.E. (1969) Hopf Algebras. Benjamin, New York. [14] Joyal, A. and Street, R. (1993) Braided Tensor Category. Advances in Mathematics, 102, 20-78. https://doi.org/10.1006/aima.1993.1055 [15] Radford, D.E. (1985) The Structure of a Hopf Algebra with a Projection. Journal of Algebra, 92, 332-347. https://doi.org/10.1016/0021-8693(85)90124-3 [16] Kassel, C. (1995) Quantum Groups. GTM 155. Springer-Verlag, Berlin. https://doi.org/10.1007/978-1-4612-0783-2 [17] Radford, D.E. and Towber, J. (1993) Yetter-Drinfeld Categories Associated to an Arbitrary Bialgebra. Journal of Pure and Applied Algebra, 87, 259-279. https://doi.org/10.1016/0022-4049(93)90114-9 [18] Zhang, S.C. and Zhang, T. (2007) Double Bicrosssum of Braided Lie Algebras. Cor- nell University, New York.

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