A Duality Theorem for L-R Crossed Product

Transcription

1 Flomat 30:5 (2016), DOI /FIL C Publshed by Faculty of Scences and Mathematcs, Unversty of Nš, Serba Avalable at: A Dualty Theorem for L-R Crossed Product Quan-guo Chen a, Dng-guo Wang b a School of Mathematcs and Statstcs, Yl Normal Unversty, Ynng, Xnjang , P.R.Chna b School of Mathematcal Scences, Qufu Normal Unversty, Qufu, Shandong , P.R.Chna Abstract. In ths work, the noton of an L-R crossed product s ntroduced as a unfed approach for L-R smash product and crossed product. Then the dualty theorem for L-R crossed product s gven. As the applcatons of the man result, some classcal results n some materals can be obtaned. 1. Introducton Classcal dualty theorems orgn n operator algebras, n works of Takesak and colaborators for descrbng the dualty between actons and coactons of locally compact groups on Von Neumann algebras ([1]). In [2], Cohen and Montgomery consdered ths dualty for actons and coactons of groups on algebras and proved that, gven a fnte group G actng as lnear automorphsms on A, there exsts an somorphsm between the smash product A G k[g] of the skew group rng A G and the dual group rng k[g] = Hom(kG, k) and the full matrx rng M n (A) over A. Ths knd of result s mportant, snce coactons of group algebras correspond to group gradngs on algebras. The extenson of ths dualty theorem to the context of Hopf algebras was made n the work of Blattner and Montgomery (see [3]). As the generalzaton of Blattner- Mongomery s result, Koppnen prove the dualty theorem for Hopf crossed product whch generalzed most of dualty theorems n [5]. From the perspectve of dualty, Wang consdered the dualty theorems of both Hopf comodule coalgebras and crossed coproducts n [6, 7]. Recently, a great deal of work has been done on the dualty theorem n [9 11] and [12]. Based on the theory of deformaton, the L-R smash product was ntroduced and studed n [13, 14]. It s defned as follows: f H s a cocommutatve balgebra and A s an H-bmodule algebra, then the L-R smash product A H s an assocatve algebra defned on A H by the multplcaton rule (a h)(b ) = (a 1 )(h 1 b) h 2 2 for any a, b A and, h H. If we replace the above multplcaton by (a h)(b ) = (a 2 )(h 1 b) h 2 1, then ths multplcaton s assocatve n [15] wthout the assumpton that H s cocommutatve. In [16], the authors ntroduced and studed the more general verson of L-R smash products Mathematcs Subject Classfcaton. Prmary 16W30 Keywords. Hopf algebra, L-R crossed product, dualty theorem Receved: 7 Aprl 2014; Accepted: 7 November 2014 Communcated by Dragana Cvetkovć Ilć Research supported by the Natonal Natural Scence Foundaton of Chna (No and ) and the Foundaton for Excelent Youth Scence and Technology Innovaton Talents of Xn Jang Uygur Autonomous Regon(No ) Emal addresses: cqg211@163.com (Quan-guo Chen), dgwang@mal.qfnu.edu.cn, dngguo95@126.com (Dng-guo Wang)

2 Q.-G. Chen, D.-G. Wang / Flomat 30:5 (2016), Followng the current trends of further research on ths topc and at the angle of unty, the paper wll present a general verson of dualty theorem for L-R crossed product whch covers most of the classcal product algebras such as smash products, crossed products and L-R smash products etc. It s the motvaton of ths paper. The paper s organzed as follows. In Secton 2, we recall some useful concepts. In Secton 3, the condtons on cocycles are establshed n order to construct L-R crossed products. Then the dualty theorem for L-R crossed product s gven n Secton 4. In Secton 5, we apply our man result to some classcal cases. 2. Prelmnares Throughout the paper, we always work over a fxed feld k and follow the Montgomery s book([17]) for termnologes on coalgebras, comodules and balgebras. Gven a vector space M, ι : M M denotes the dentty map. Recall that a left (rght) measure of H on an algebra A s a lnear map H A A(A H A) gven by h a h a(a h) = a h) such that, for any h H, a, b A, h (ab) = (h 1 a)(h 2 b)(resp.(ab) h = (a h 1 )(b h 2 )), h 1 A = ε H (h)1 A, 1 H a = a (resp. 1 A h = ε H (h)1 A, a 1 H = a). Gven a left (rght) measure of H on A, f the measure s module acton, then we can get the left (rght)- module algebra. If an algebra A s both a left H-module algebra and a rght H-module algebra wth the compatble module actons, then A s called an H-bmodule algebra. 3. L-R Crossed Products In ths secton, we shall ntroduce the noton of a L-R crossed product. Assume that H measures on A from the left. Let A be a rght H-module algebra wth the compatblty wth the left measure, and σ : H H A a lnear map. Defne a multplcaton on vector space A H by for any a, b A and h, l H. (a h)(b ) = (a 3 )(h 1 b)σ(h 2, 1 ) h 3 2 Defnton 3.1. Let H be a Hopf algebra, A a rght H-module algebra and σ : H H A a lnear map. We say that H s σ-cocommutatve, f the followng relaton holds, for all l,, h H. σ(l, ) h 1 h 2 = σ(l, ) h 2 h 1 Remark 3.2. If σ s trval,.e., σ(h, ) = ε H (h)ε H ( )1 A. Then H s σ-cocommutatve. The followng theorem gves the necessary and suffcent condtons under whch A H s assocatve and A H s untal wth 1 A 1 H as the dentty element. Theorem 3.3. Assume that H measures on A from the left. Let A be a rght H-module algebra wth the compatblty wth the left measure, and σ : H H A a lnear map such that H s σ-cocommutatve. Then () 1 A 1 H s the unt of A H f and only f, for all a A, σ(h, 1 H ) = ε H (h)1 A = σ(1 H, h), (3.1) () A H s assocatve f and only f the followng condtons hold: (h 1 σ(l 1, m 1 ))σ(h 2, l 2 m 2 ) = (σ(h 1, l 1 ) m 1 )σ(h 2 l 2, m 2 ), (3.2) (h 1 (l 1 a))σ(h 2, l 2 ) = σ(h 1, l 1 )(h 2 l 2 a) (3.3) for any h, l, m H and a A.

3 Q.-G. Chen, D.-G. Wang / Flomat 30:5 (2016), Proof. The proof of () s straghtforward, so we omt t. Now, we shall check (). Suppose A H s assocatve, we have and So t follows that (1 A h)[(1 A l)(a m)] = (1 A h)[(l 1 a)σ(l 2, m 1 ) l 3 m 2 ] = (h 1 ((l 1 a)σ(l 2, m 1 )))σ(h 2, l 3 m 2 ) h 3 l 4 m 3 [(1 A h)(1 A l)](a m) = (σ(h 1, l 1 ) h 2 l 2 )(a m) = (σ(h 1, l 1 ) m 3 )(h 2 l 2 a)σ(h 3 l 3, m 1 ) h 4 l 4 m 2. (h 1 ((l 1 a)σ(l 2, m 1 )))σ(h 2, l 3 m 2 ) h 3 l 4 m 3 = (σ(h 1, l 1 ) m 3 )(h 2 l 2 a)σ(h 3 l 3, m 1 ) h 4 l 4 m 2. Applyng ι ε H to both sde of the above equalty, we have (h 1 ((l 1 a)σ(l 2, m 1 )))σ(h 2, l 3 m 2 ) = (σ(h 1, l 1 ) m 2 )(h 2 l 2 a)σ(h 3 l 3, m 1 ). (3.4) If we take a = 1 A n (3.4) and use that H s σ-cocommutatve, we get (h 1 σ(l 1, m 1 ))σ(h 2, l 2 m 2 ) = (σ(h 1, l 1 ) m 1 )σ(h 2 l 2, m 2 ). (3.5) If we take m = 1 H n (3.4), t follows that (h 1 (l 1 a))σ(h 2, l 2 ) = σ(h 1, l 1 )(h 2 l 2 a). (3.6) Conversely, assume that (3.2) and (3.3) hold. Frst, we need the followng equalty (h 1 (l 1 a))(σ(h 2, l 2 ) m) = (σ(h 1, l 1 ) m)(h 2 l 2 a)). (3.7) As a matter of fact, for all h, l, m H and a A, we have (h 1 (l 1 a))(σ(h 2, l 2 ) m) = ((h 1 (l 1 a s(m 1 )))σ(h 2, l 2 )) m 2 (3.3) = (σ(h 1, l 1 )(h 2 l 2 (a s(m 1 )))) m 2 = (σ(h 1, l 1 ) m 2 )(h 2 l 2 (a s(m 1 )m 3 )) = (σ(h 1, l 1 ) m 3 )(h 2 l 2 (a s(m 1 )m 2 )) = (σ(h 1, l 1 ) m)(h 2 l 2 a). Then, for all a, b, c A and h, l, m H, we have (a h)[(b l)(c m)] = (a h)[(b m 3 )(l 1 c)σ(l 2, m 1 ) l 3 m 2 ] = (a l 5 m 4 )(h 1 ((b m 5 )(l 1 c)σ(l 2, m 1 )))σ(h 2, l 3 m 2 ) h 3 l 4 m 3 = (a l 5 m 4 )(h 1 (b m 5 ))(h 2 (l 1 c))(h 3 σ(l 2, m 1 ))σ(h 4, l 3 m 2 ) h 5 l 4 m 3 = (a l 5 m 4 )(h 1 (b m 5 ))(h 2 (l 1 c))(σ(h 3, l 2 ) m 2 )σ(h 4 l 3, m 1 ) h 5 l 4 m 3

4 Q.-G. Chen, D.-G. Wang / Flomat 30:5 (2016), (3.7) = (a l 5 m 4 )((h 1 b) m 5 )(σ(h 2, l 1 ) m 2 }{{} )(h 3l 2 c)σ(h 4 l 3, m 1 ) h 5 l 4 m 3 }{{} = (a l 5 m 4 }{{} )((h 1 b) m 5 )(σ(h 2, l 1 ) m 3 }{{} )(h 3l 2 c)σ(h 4 l 3, m 1 ) h 5 l 4 m 2 = (a l 5 m 3 )((h 1 b) m 5 }{{} )(σ(h 2, l 1 ) m 4 }{{} )(h 3l 2 c)σ(h 4 l 3, m 1 ) h 5 l 4 m 2 = (a l 5 m 3 )((h 1 b) m 4 )(σ(h 2, l 1 ) m 5 )(h 3 l 2 c)σ(h 4 l 3, m 1 ) h 5 l 4 m 2 = (((a l 5 )(h 1 b)σ(h 2, l 1 )) m 3 )(h 3 l 2 c)σ(h 4 l 3, m 1 ) h 5 l 4 m 2 = ((a l 3 )(h 1 b)σ(h 2, l 1 ) h 3 l 2 )(c m) = [(a h)(b l)](c m). Ths ends the proof. We call the k-algebra A H an L-R crossed product, denoted by A σ H. Example 3.4. Consder the group algebra kz wth the obvous Hopf algebra structure and let be a generator of Z n multplcaton notaton. Fx an element 0 q k, and defne a lnear map σ : kz kz kz, j q j 1 and two actons on kz: t l = q tl l, t l = q tl t. Snce and ( t l ) k = q tl l k = q tl kl l t ( l k ) = q lk t l = q tl lk l, t follows that (kz,, ) s kz-bmodule. It s not hard to show that (kz, ) s a left kz-module algebra and (kz, ) s a rght kz-module algebra. Straghtforward computaton can show that σ s a cocycle and condtons (3.2) and (3.3) hold. Thus we have the L-R crossed product kz σ kz wth the multplcaton va ( m l )( n t ) = q nl+lt mt m+n l+t. Example 3.5. Consder the polynomal algebra k[x] wth the coalgebra structure and the antpode gven by (X n ) = n C k nx k X n k, ε(x n ) = 0, S(X n ) = ( 1) n X n, n > 0. k=0 Fx an element 0 q k, and defne a lnear map σ : k[x] k[x] k[x] va { 0, f j; σ(x, X j ) =!q 1, f = j. Two actons of k[x] on k[x] are gven by { 0, f > j; X X j = j! (j )! q X j, f j, { 0, f > j; X j X = ( 1) j! (j )! q X j, f j. It s not hard to show that (k[x],, ) s k[x]-bmodule, (k[x], ) s a left k[x]-module algebra and (k[x], ) s a rght k[x]-module algebra. Snce { 0, f 0; σ(x, 1) = 1, f = 0, t follows that σ(x, 1) = ε(x )1. Smlarly, we can check that σ(1, X ) = ε(x )1. Straghtforward computaton can show that the condtons (3.2) and (3.3) hold. Thus, we have another L-R crossed product k[x] σ k[x].

5 4. The Dualty Theorem for L-R-Crossed product Q.-G. Chen, D.-G. Wang / Flomat 30:5 (2016), Let A be a rght H-module algebra. Assume that there exsts a left measure of H on A such that H s σ-cocommutatve. If H s a fnte dmensonal Hopf algebra, the dual vector space H has a natural structure of a Hopf algebra. Now, we wll construct the dualty theorem for an L-R crossed product. Frst, we need some lemmas. Lemma 4.1. Let H be a fnte dmensonal Hopf algebra. Then A σ H s a left H -module algebra va for any a A, h H and f H. Lemma 4.2. The map f (a σ h) = a σ h 1 f (h 2 ) ϕ : (A σ H) H End(A σ H) A (here means smash product and End(A σ H) A means the rght A-module endomorphsms) defned by ϕ((a σ h) f )(b σ ) = (a σ h)(b σ 1 ) f ( 2 ) for any a, b A, h, H and f H, s a homomorphsm of algebras, where A σ H s a rght A-module va (a σ h) b = (a σ h)(b σ 1 H ). Proof. Frst, we wll show that ϕ commutes wth the rght acton of A on A σ H. Indeed, for any a, b, d A, h, H and f H, we compute ϕ((a σ h) f )((b σ ) d) = ϕ((a σ h) f )(b( 1 d) σ 2 ) = (a 4 )(h 1 (b( 1 d)))σ(h 2, 2 ) σ h 3 3 f ( 5 ) = (a 4 )(h 1 b)(h 2 ( 1 d))σ(h 3, 2 ) σ h 4 3 f ( 5 ) (3.3) = (a 4 )(h 1 b)σ(h 2, 1 )(h 3 2 d) σ h 4 3 f ( 5 ) = ((a 3 )(h 1 b)σ(h 2, 1 ) σ h 3 2 f ( 4 )) d = (ϕ((a σ h) f )(b σ )) d. Next, for all a, b, x A, h, l, y H and f, H, we have ϕ((a σ h) f ) ϕ((b σ l) )(x σ y) = ϕ((a σ h) f )((b y 3 )(l 1 x)σ(l 2, y 1 ) σ l 3 y 2 ) (y 4 ) = (a σ h)(b y 4 )(l 1 x)σ(l 2, y 1 ) σ l 3 y 2 (y 5 ) f (l 4 y 3 ) = (a l 5 y 4 )(h 1 ((b y 6 )(l 1 x)σ(l 2, y 1 )))σ(h 2, l 3 y 2 ) σ h 3 l 4 y 3 (y 7 ) f (l 6 y 5 ) = (a l 5 y 4 )(h 1 (b y 6 ))(h 2 (l 1 x)) (h 3 σ(l 2, y 1 ))σ(h 4, l 3 y 2 ) } {{ } σh 5 l 4 y 3 (y 7 ) f (l 6 y 5 ) (3.2) = (a l 5 y 4 )(h 1 (b y 6 )) (h 2 (l 1 x))(σ(h 3, l 2 ) y 2 ) } {{ } σ(h 4l 3, y 1 ) σ h 5 l 4 y 3 (y 7 ) f (l 6 y 5 ) (3.3) = (a l 5 y 4 )(h 1 (b y 6 ))(σ(h 2, l 1 ) y 2 )(h 3 l 2 x)σ(h 4 l 3, y 1 ) σ h 5 l 4 y 3 (y 7 ) f (l 6 y 5 ) = (a l 5 y 4 )(h 1 (b y 5 ))(σ(h 2, l 1 ) y 2 )(h 3 l 2 x)σ(h 4 l 3, y 1 ) σ h 5 l 4 y 3 (y 7 ) f (l 6 y 6 ) = (a l 5 y 4 )(h 1 (b y 5 ))(σ(h 2, l 1 ) y 3 )(h 3 l 2 x)σ(h 4 l 3, y 1 ) σ h 5 l 4 y 2 (y 7 ) f (l 6 y 6 ) = (a l 5 y 3 )(h 1 (b y 5 ))(σ(h 2, l 1 ) y 4 )(h 3 l 2 x)σ(h 4 l 3, y 1 ) σ h 5 l 4 y 2 (y 7 ) f (l 6 y 6 ) = (a l 5 y 3 )((h 1 b) y 4 ))(σ(h 2, l 1 ) y 5 )(h 3 l 2 x)σ(h 4 l 3, y 1 ) σ h 5 l 4 y 2 (y 7 ) f (l 6 y 6 ) = (((a l 5 )(h 1 b)(σ(h 2, l 1 ))) y 3 )(h 3 l 2 x)σ(h 4 l 3, y 1 ) σ h 5 l 4 y 2 (y 5 ) f (l 6 y 4 ) = ϕ(((a σ h) f )(b σ l) )(x σ y).

6 Q.-G. Chen, D.-G. Wang / Flomat 30:5 (2016), Ths ends the proof. Lemma 4.3. Let H be a fnte dmensonal Hopf algebra and A σ H be the L-R crossed product wth convoluton nverse σ. Then (σ 1 (h 1, l 1 ) m)(h 2 (l 2 a)) = (h 1 l 1 a)(σ 1 (h 2, l 2 ) m), (4.1) σ 1 (l, ) h 1 h 2 = σ 1 (l, ) h 2 h 1, σ(h 1 l 1, m 1 )σ 1 (h 2, l 2 m 2 ) = (σ 1 (h 1, l 1 ) m 1 )(h 2 σ(l 2, m 2 )). (4.2) Proof. Here we only check that (4.2) holds. Multplyng convolutvely on the rght of (3.2) by σ 1, we have Ths gves Snce (h 1 σ(l 1, m 1 ))σ(h 2, l 2 m 2 )σ 1 (h 3, l 3 m 3 ) = (σ(h 1, l 1 ) m 1 )σ(h 2 l 2, m 2 )σ 1 (h 3, l 3 m 3 ). h σ(l, m) = (σ(h 1, l 1 ) m 1 )σ(h 2 l 2, m 2 )σ 1 (h 3, l 3 m 3 ). (4.3) (σ 1 (h 1, l 1 ) m 1 )(h 2 σ(l 2, m 2 )) (4.3) = (σ 1 (h 1, l 1 ) m 1 )(σ(h 2, l 2 ) m 2 )σ(h 3 l 3, m 3 )σ 1 (h 4, l 4 m 4 ) = σ(h 1 l 1, m 1 )σ 1 (h 2, l 2 m 2 ), t follows that (4.2) holds. Let {e } be a bass of H and {e } be the dual bass of H,.e., such that e (e j) = δ j for all, j. Then we have the followng denttes: e (h)e = h, e f (e ) = f for all h H and f H. Lemma 4.4. Let H be a fnte dmensonal Hopf algebra and A σ H be the L-R crossed product wth convoluton nverse σ. Defne a lnear map ψ : End(A σ H) A (A σ H) H by Then the maps ϕ and ψ are nverse of each other. Proof. We need to check that For all a A, h H and f H, we have ψ : T (T(σ 1 (e 4, S 1 (e 3 )) e 2 σ e 5 )(1 A σ S 1 (e 1 ))) e. ϕ ψ = ι, ψ ϕ = ι. ψ ϕ((a σ h) f ) = [(a σ h)(σ 1 (e 4, S 1 (e 3 )) e 2 σ e 5 )(1 A σ S 1 (e 1 ))] e f (e 6) = [(a σ h)((σ 1 (e 6, S 1 (e 5 )) e 4 S 1 (e 1 ))σ(e 7, S 1 (e 3 ))) σ e 8 S 1 (e 2 )] e f (e 9) = [(a e 12 S 1 (e 2 ))(h 1 (σ 1 (e 8, S 1 (e 7 )) e 6 } {{ } S 1 (e 1 )σ(e 9, S 1 (e 5 ) } {{ } ))) σ(h 2, e 10 S 1 (e 4 )) σ h 3 e 11 S 1 (e 3 )] e f (e 13)

7 Q.-G. Chen, D.-G. Wang / Flomat 30:5 (2016), = [(a e 12 S 1 (e 2 ))(h 1 ((σ 1 (e 8, S 1 (e 7 )) e 5 } {{ } S 1 (e 1 )σ(e 9, S 1 (e 6 )))) σ(h 2, e 10 S 1 (e 4 ) } {{ } ) σh 3 e 11 S 1 (e 3 )] e f (e 13) = [(a e 12 S 1 (e 2 ))(h 1 (σ 1 (e 8, S 1 (e 7 )) e 4 } {{ } S 1 (e 1 )σ(e 9, S 1 (e 6 ))) σ(h 2, e 10 S 1 (e 5 )) σ h 3 e 11 S 1 (e 3 )] } {{ } e f (e 13) = [(a e 12 S 1 (e 2 } {{ } ))(h 1 (σ 1 (e 8, S 1 (e 7 )) e 3 } {{ } S 1 (e 1 )σ(e 9, S 1 (e 6 )))) σ(h 2, e 10 S 1 (e 5 )) σ h 3 e 11 S 1 (e 4 )] e f (e 13) = [(a e 12 S 1 (e 3 ))(h 1 (σ 1 (e 8, S 1 (e 7 )) e 2 S 1 (e 1 ) } {{ } σ(e 9, S 1 (e 6 )))) σ(h 2, e 10 S 1 (e 5 )) σ h 3 e 11 S 1 (e 4 )] e f (e 13) = [(a e 10 S 1 (e 1 ))(h 1 (σ 1 (e 6, S 1 (e 5 ))σ(e 7, S 1 (e 4 ) } {{ } ))) σ(h 2, e 8 S 1 (e 3 )) σ h 3 e 9 S 1 (e 2 )] e f (e 11) = [(a e 10 S 1 (e 1 ))(h 1 (σ 1 (e 6, S 1 (e 5 ))σ(e 7, S 1 (e 4 )))) σ(h 2, e 8 S 1 (e 3 )) σ h 3 e 9 S 1 (e 2 )] e f (e 11) = ((a e 6 S 1 (e 1 ))σ(h 1, e 4 S 1 (e 3 ) } {{ } ) σh 2 e 5 S 1 (e 2 )) e f (e 7) = ((a e 4 S 1 (e 1 ) } {{ } ) σh e 3 S 1 (e 2 } {{ } )) e f (e 5) = (a σ h) e f (e ) = (a σ h) f. So we get ψ ϕ = ι. As to ϕ ψ = ι, we proceed the proof as follows: ϕ ψ(t)(a σ h) = ϕ((t(σ 1 (e 4, S 1 (e 3 ) e 2 σ e 5 ))(1 A σ S 1 (e 1 ))) e )(a σh) = T(σ 1 (e 4, S 1 (e 3 )) e 2 σ e 5 )(1 A σ S 1 (e 1 ))(a σ h 1 )e (h 2) = T(σ 1 (h 8, S 1 (h 7 )) h 6 σ h 9 )((S 1 (h 5 ) a)σ(s 1 (h 4 ), h 1 ) σ S 1 (h 3 )h 2 } {{ } ) = T(σ 1 (h 6, S 1 (h 5 )) h 4 σ h 7 )((S 1 (h 3 ) a)σ(s 1 (h 2 ), h 1 ) σ 1 H ) = T((σ 1 (h 6, S 1 (h 5 )) h 4 σ h 7 )((S 1 (h 3 ) a)σ(s 1 (h 2 ), h 1 ) σ 1 H )) = T((σ 1 (h 6, S 1 (h 5 )) h 4 } {{ } )(h 7 (S 1 (h 3 ) } {{ } a))(h 8 σ(s 1 (h 2 ), h 1 )) σ h 8 )) = T((σ 1 (h 6, S 1 (h 5 )) h 3 )(h 7 (S 1 (h 4 ) a))(h 8 σ(s 1 (h 2 ), h 1 )) σ h 8 )) (4.1) = T((h 6 S 1 (h 5 ) } {{ } a)(σ 1 (h 7 S 1 (h 4 )) h 3 ))(h 8 σ(s 1 (h 2 ), h 1 )) σ h 9 )) = T(a(σ 1 (h 5 S 1 (h 4 )) h 3 ))(h 6 σ(s 1 (h 2 ), h 1 )) σ h 7 ))

8 Q.-G. Chen, D.-G. Wang / Flomat 30:5 (2016), = T(a(σ 1 (h 5 S 1 (h 4 )) h 2 ))(h 6 σ(s 1 (h 3 ), h 1 )) σ h 7 )) (4.2) = T(aσ(h 5 S 1 (h 4 ) } {{ }, h 1)σ 1 (h 6 S 1 (h 3 ) } {{ }, h 2) σ h 7 }{{} ) = T(a σ h). Ths proof s completed. From the lemmas above, we can get the followng man result n ths secton. Theorem 4.5. Let H be a fnte dmensonal Hopf algebra and A σ H be the L-R crossed product wth convoluton nverse σ such that H s σ-cocommutatve. Then there s a canoncal somorphsm between the algebras (A σ H) H and End(A σ H) A. 5. Applcatons In ths secton, we shall gve some applcatons of Theorem 4.5, some classcal results n several materals can be obtaned Crossed Products If the rght H-module acton of A s trval, that s, a h = aε H (t) for any a A and h H, then A s an H-bmodule and (3.2) holds, and A σ H recovers to the usual crossed product n sense of [4]. From Theorem 4.5, we have Corollary 5.1. ([5]) Let H be a fnte dmensonal Hopf algebra and A σ H be the usual crossed product wth convoluton nverse σ. Then there s a canoncal somorphsm between the algebras (A σ H) H and End(A σ H) A L-R Smash Products If σ s trval, that s, σ(h, ) = ε H (h)ε H ( )1 A, then A σ H reduces to the usual L-R smash product. From Theorem 4.5, we have Corollary 5.2. ([12]) Let H be a fnte dmensonal Hopf algebra and A H be the usual L-R smash product. Then there s a canoncal somorphsm between the algebras (A H) H and End(A H) A. Furthermore, f the rght H-module acton of A s trval, then L-R smash product A H s exactly the usual smash product. From Corollary 5.2, we have Corollary 5.3. ([3]) Let H be a fnte dmensonal Hopf algebra and A H be the usual smash product. Then there s a canoncal somorphsm between the algebras (A H) H and End(A H). References [1] Y. Nakagam, M. Takesak: Dualty for crossed products of Von Neumann algebras, Lecture Notes n Math. 731, Sprnger Verlag (1979). [2] M. Cohen, S. Montgomery: Group-graded rngs, smash products, and group actons, Trans. Amer. Math. Soc., 282(1984), [3] R. J. Blattner, S. Montgomery: A dualty theorem for Hopf module algebras, J. Algebra, 95(1985), [4] R. Blattner, M. Cohen, S. Montgomery: Crossed products and nner actons of Hopf algebras, Trans. Amer. Math. Soc., 298(1986), [5] M. Koppnen: A dualty theorem for crossed products of Hopf algebras, J. Algebra, 146(1992), [6] S. H. Wang: A dualty theorem for Hopf comodule coalgebra, Chnese Scence Bulletn, 39(1994), [7] S. H. Wang: A dualty theorem for crossed coproduct for Hopf algebras, Scence n Chna, 38(1995), 1-7. [8] C. Lomp: Dualty for Partal Group Actons, Int. Electron. J. Algebra, 4(2008), [9] B. L. Shen: Maschke-type theorem, Dualty theorem, and the global dmenson for weak crossed products, Comm. Algebra, 40(2012), [10] B. L. Shen, S. H. Wang: On group crossed coproduct, Int. Electron. J. Algebra, 4(2008), [11] B. L. Shen, S. H. Wang: Blattner-Cohen-Montgomery s Dualty Theorem for (Weak) Group Smash Products, Comm. Algebra, Comm. Algebra, 36(2008),

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