Advances in Pure Mathematics, 2014, 4, 522-528 Published Online September 2014 in SciRes. http://www.scirp.org/journal/apm http://dx.doi.org/10.4236/apm.2014.49060 The Freedom of Yetter-Drinfeld Hopf Algebras Yanhua Wang School of Mathematics, Shanghai University of Finance and Economics, Shanghai, China Email: [email protected] Received 1 August 2014; revised 2 September 2014; accepted 13 September 2014 Copyright © 2014 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract In this paper, the fundamental theorem of Yetter-Drinfeld Hopf module is proved. As applications, the freedom of tensor and twisted tensor of two Yetter-Drinfeld Hopf algebras is given. Let A be a Yetter-Drinfeld Hopf algebra. It is proved that the category of A-bimodule is equivalent to the cat- egory of AA⊗ -twisted module. Keywords Hopf Algebra, Hopf Module, Yetter-Drinfeld Module, Yetter-Drinfeld Hopf Algebra 1. Introduction Let k be a field and A an algebra. A left A -module is a k -vector space V together with a k -linear map ⊗→ such that →=→ → and →=. The category of left -module is denoted by AV V abvabv( ) 1 vv A ∆ ε A M . Dually, let (C,,) be a coalgebra. A left C -comodule is a k -vector space V together with a k -linear map ρ :V→⊗ VC such that −10 a−−1⊗⊗ a 1 a 0= a − 1⊗ a 0⊗ a 0, ε aa−10= a. ∑∑( )12( ) ( ) ( ) ∑( ) C The category of left C -comodule is denoted by M . For more about modules and comodules, see [1]-[3]. Assume that H is a Hopf algebra with antipode S , a left Yetter-Drinfeld module over H is a k -vector space V which is both a left H -module and left H -comodule and satisfies the compatibility condition −10−10 ∑∑(hv→=) ⊗ ( hv→⊗) hvShh1( 32) → v, H for all h∈∈ Hv, V. The category of left Yetter-Drinfeld module is denoted by H YD . Yetter-Drinfeld mod- ules category constitutes a monomidal category, see [4]. The category is pre-braided; the pre-braiding is given by How to cite this paper: Wang, Y.H. (2014) The Freedom of Yetter-Drinfeld Hopf Algebras. Advances in Pure Mathematics, 4, 522-528. http://dx.doi.org/10.4236/apm.2014.49060 Y. H. Wang −10 τVW, :VW⊗→⊗ WV, vw ⊗ ( v→ w) ⊗ v. The map is a braiding in H YD precisely when Hopf algebra H has a bijective antipode S with inverse H τ S of S . In this case, the inverse of VW, is −−1 01 τVW, :W⊗→⊗ V V W, w⊗ v ∑ v⊗ Sv( ) → w . H Let H be a Hopf algebra and H YD the category of left Yetter-Drinfeld module over H . We call A a H Hopf algebra in H YD or Yetter-Drinfeld Hopf algebra if A is a k -algebra and a k -coalgebra, and the fol- lowing conditions (a1)-(a6) hold for h∈∈ H, ab, A, (a1) A is a left H -module algebra, i.e., h→=( ab) ∑( h12→ a)( h→ b), h→1AA=ε ( h)1 . (a2) A is a left H -comodule algebra, i.e., −10−−1 1 00 ρρ(ab) = ∑∑( ab) ⊗⊗( ab) = a b a b , (1A) = 1 HA⊗ 1 . (a3) A is a left H -module coalgebra, i.e., ∆→(ha→=) ∑( h11→ a) ⊗( h 2 a 2), ε( ha→=) εεHA( h) ( a). (a4) A is a left H -comodule coalgebra, i.e., a−1⊗ a 0 ⊗ a 0 = aa−−11⊗⊗ a 0 a 0, a− 1ε a 0= ε ( a)1 . ∑∑( )12( ) 12 1 2 ∑ A( ) AH H (a5) ∆, ε are algebra maps in H YD , i.e., −10 ∆(ab) =∑ a1( a 2→∆ b 1) ⊗ a 22 b , (1) =⊗= 1 1, ε(ab) εε( a) ( b), ε(1Ak) = 1. H (a6) There exists a k -linear map SA: → A in H YD such that ∑∑Sa( 12) a=ε ( a)1A = aSa1( 2) One easily get that S is both H -linear and H -colinear. In general, Yetter-Drinfeld Hopf algebras are not ordinary Hopf algebras because the bialgebra axiom asserts that they obey (a5). However, it may happen that Yetter-Drinfeld Hopf algebras are ordinary Hopf algebras when the pre-braiding is trivial, for details see [5]. Yetter-Drinfeld Hopf algebras are generalizations of Hopf algebras. Some important properties of Hopf alge- bras can be applied to Yetter-Drinfeld Hopf algebra. For example: Doi gave the trace formular of Yetter-Drin- feld Hopf algebras in [6] and studied Hopf module in [7]; Chen and Zhang constructed Four-dimensional Yet- ter-Drinfeld module algebras in [8]; Zhu and Chen studied Yetter-Drinfeld modules over the Hopf-Ore Exten- sion of Group algebra of Dihedral group in [9]; Alonso Álvarez, Fernández Vilaboa, González Rodríguez and Soneira Calvoar considered Yetter-Drinfeld modules over a weak braided Hopf algebra in [10], and so on. Hopf module fundamental theorem plays an important role in Hopf algebras. This theory can be generalized to Yetter-Drinfel Hopf algebras. A Theorem 1.1. Let A be a Yetter-Drinfeld Hopf algebra, MM∈A be a Yetter-Drinfeld Hopf module, then MA≅⊗coA M as left A Yetter-Drinfeld Hopf module. Note that Theorem 1.1 was appeared in [7], but we give a different proof with Doi’s here. τ Let A be a Yetter-Drinfeld Hopf algebra. Define the multiplication of ( AA⊗ ) as (a⊗ b)( c ⊗= d) ∑ ab( −10→ c) ⊗ bd, τ then ( AA⊗ ) is an algebra. But it is not a Yetter-Drinfeld Hopf algebra if τ is not the trivial twist T. As ap- plications of Yetter-Drinfeld Hopf module fundamental theory, we have the freedom of the tensor of Yet- ter-Drinfeld Hopf algebras and twisted tensor of Yetter-Drinfeld Hopf algebras. τ Theorem 1.2. Let A be a Yetter-Drinfeld Hopf algebra, then AA⊗ and ( AA⊗ ) are free over A . 523 Y. H. Wang τ We also proved the category of Yetter-Drinfeld. A -bimodule is equivalent to the category of ( AA⊗ ) - module. Theorem 1.3. Let A be a Yetter-Drinfeld Hopf algebra. Then the category of M and τ M are AA ( AA⊗ ) equivalent. 2. The Freedom of Yetter-Drinfeld Hopf Algebras In this section, we require H is a Hopf algebra and A is a Yetter-Drinfeld module over H . Moreover, we need A is a Yetter-Drinfeld Hopf algebra. Next, we will give the definition of Yetter-Drinfeld Hopf module, also see [7]. Definition 2.1. Let A be a Yetter-Drinfeld Hopf algebra. The Yetter-Drinfeld Hopf module over A is de- fined by the following 1) M is a left A -module and left A -comodule with comodule map ρM : M→⊗ AM, −12 2) ρM is a A -module map, i.e., ρM (am) = ∑ a1( a22→ m− 10) ⊗ a m , where aa12, ∈ A. −10 Note that A is a left H -comodule with ρ A (a) = ∑ a⊗ aaA, ∈ , and M is a left A -comodule with A ρM (m) = ∑ m−10⊗ mmM, ∈ . The Yetter-Drinfeld Hopf module category over A is denoted by A M . coA Define Mm={ ρM ( m) =1 ⊗ m} is the set of coinvariant elelments of M . Next conclusion is similar to the fundamental theorem of Hopf algebra, we call it as the fundamental theorem of Yetter-Drinfeld Hopf mod- ule. A Theorem 2.2. Let A be a Yetter-Drinfeld Hopf algebra, MM∈A be a Yetter-Drinfeld Hopf module. Then MA≅⊗coA M as left A Yetter-Drinfeld Hopf module. Proof: We define α : A⊗coA MM → by a⊗⋅ m am and β : MA→⊗coA M by m ∑ m−−2⊗ Sm( 10)⋅ m. coA First, we show that β is well-defined, i.e., ∑ Sm( −10)⋅∈ m M. In fact, we have −1 0 ρ Sm⋅= m Sm Sm→⊗ m Sm⋅ m (∑∑( −10) ) ( ( −2))12( ( −− 2)2 1) ( ( − 2)) 0 −−2 1 0 = ∑ m−3→ Sm( −− 23)( m→⊗ m − 1) Sm( − 30) ⋅ m 0 = ∑ m−−3→⊗( Sm( 21)( m− )) Sm( − 30) ⋅ m 0 = ∑ m−−21→⊗ε ( m) Sm( −2 ) ⋅ m 0 =∑ Sm( −10) ⋅ m. coA So ∑ Sm( −10) m∈ M. Thus β is well-defined. We will show that α is the inverse of β . Indeed, if mM∈ we have αβ(m) = α (∑ m−2⊗ Sm( − 10)⋅ m) = ∑∑ mSm−− 2( 10) ⋅= mε ( m− 10) m= m, Hence αβ = id . Conversely, if m∈∈coA Ma, A, then βα⊗⊗= β ⋅= ⋅ ⋅ ⋅⋅ (a m) ( am) ∑( am)−−2 Sam( ) 10( am) −−1 12 = ∑ aa11110( 22→ m−−) ⊗ S aa( → m) a2 ⋅ m ( )1 (( )2 ) = ∑ a12⊗⊗ Sa( ) a3 ⋅ m= a m, which show that βα = id too. It remains to show that α is a morphism of H -module and H -comodule. The first assertion is clear, since αα(bam⋅⊗( )) =( bambamb ⊗) = ⋅=⋅ α( am ⊗). Next, we show that α is a H -comodule morphism, i.e. ρα=(id ⊗ α )( ∆⊗ id ) . Indeed, we have 524 Y. H. Wang −1 0 ρα(a⊗ m) = ρ ( a ⋅ m) =∑∑ a12( a→⊗ m− 1) a2 ⋅ m 0 = a12 ⊗ a ⋅ m =( id ⊗α )( ∆⊗ id)( a⊗ m). This complete the proof. Proposition 2.3. We have AA⊗ is a Yetter-Drinfeld Hopf module over A . Proof: AA⊗ is an A -module by the trivial module action: abc⋅⊗=( ) abc ⊗. In fact, for ab,,∈ A cdAA⊗∈⊗ , we have a( b( c⊗ d)) = a( bc ⊗ d) = abc ⊗= d( ab)( c ⊗ d ) and 1(cd⊗=⊗) cd. The A-co- module structure of AA⊗ is defined by ρ (ab⊗) =∑ a12⊗⊗ a bAAA ∈⊗⊗. It is easy to check AA⊗ is an Yetter-Drinfeld Hopf module over A , we omit it.