The Maschke Theorem for L-R-smash Products

Transcription

1 139ò16Ï ê Æ? Ð Vol.39, No c12 ADVANCES IN MATHEMATICS Dec., 2010 The Maschke Theorem for L-R-smash Products NIU Ruifang, ZHOU Xiaoyan, WANG Yong, ZHANG Liangyun (Dept. of Math., Nanjing Agricultural University, Nanjing, Jiangsu, , P. R. China) Abstract: In this paper, we give a sufficient and necessary condition for generalized L-Rsmash products or generalized diagonal crossed products to be bialgebras(hopf algebras), and mainly give a Maschke theorem for L-R-smash products, and hence obtain Maschke theorems for twisted smash products and diagonal crossed products. Key words: bimodule algebra; bicomodule algebra; L-R-smash product; twisted smash product; diagonal crossed product MR(2000) Subject Classification: 16W30 / CLC number: O153.3 Document code: A Article ID: (2010) Introduction and Preliminaries The L-R-smash product was introduced and studied in series of papers [1 4], with motivations and examples coming from the theory of deformation quantization. It is defined as follows: if H is a bialgebra, A is an H-bimodule algebra via actions and, and the L-R-smash product A H is an associative algebra defined on A H by the multiplication rule (a h)(b g) = Σ(a g 2 )(h 1 b) h 2 g 1 for any a, b A, h, g H. If the right H-action is trivial, A H coincides with the ordinary smash product A#H. Panaite in [5] introduced a more general version of the so-called generalized L-R-smash product and studied its relations with the generalized diagonal crossed products. Zhang in [6] developed the theory of L-R-smash products and gave a sufficient and necessary condition for L-R-smash product to be a bialgebra(hopf algebra). This paper mainly develops the theory for generalized L-R-smash products or generalized diagonal crossed products and considers the Maschke theorem for the usual L-R-smash product A H, and obtains Maschke theorems for twisted smash products and diagonal crossed products. It is organized as follows. In Section 2, we give some sufficient and necessary conditions for generalized L-R-smash products and generalized diagonal crossed products to be bialgebras(hopf algebras). In Section 3, we give a new description for a Maschke theorem of L-R-smash products. We always work over a fixed field k and follow Montgomery s book in [7, 13] for terminologies on module algebras, module coalgebras, comodule algebras and comodule coalgebras. Received date: Revised date: Foundation item: Supported by EMSFC(No ) and NSFC(No ). zlyun@njau.edu.cn.

2 658 ê Æ? Ð 39ò Let A be a bialgebra. If A is both an H-bimodule algebra(that is, it is an (H, H)-bimodule, a left H-module algebra and a right H-module algebra) and an H-bimodule coalgebra(that is, it is an (H, H)-bimodule, a left H-module coalgebra and a right H-module coalgebra), then A is called an H-bimodule bialgebra. Similarly, we can define an H-bicomodule bialgebra. Let H be a Hopf algebra with antipode S, and A be an H-bimodule algebra. Then, by Proposition 1.1 in [8], we have the twisted smash product A H whose multiplication is given by (a h)(b l) = Σa(h 1 b S(h 3 )) h 2 l on the tensor space A H, for any a, b A, h, l H. Let H be a finite dimensional Hopf algebra. Define the structures as follows: h f = Σf 2 (h)f 1, f h = Σf 1 (h)f 2 for any f H, h H. Then, by Example 1.5 in [8], (H,, ) is an H-bimodule algebra and hence we obtain the twisted smash product H H whose multiplication is given by (f h)(g l) = Σf(h 1 g S(h 3 )) h 2 l. In particular, if S 2 = id, then H H is exactly the Drinfel d double D(H) given in [7]. A bialgebra H is called a Long bialgebra in [9, 10] if there exists a linear map σ : H H k, such that for any x, y, z H, (L1) Σσ(x 1, y)x 2 = Σσ(x 2, y)x 1, (L2) σ(x, 1) = ε(x), (L3) σ(x, yz) = Σσ(x 1, y)σ(x 2, z), (L4) σ(1, x) = ε(x), (L5) σ(xy, z) = Σσ(x, z 2 )σ(y, z 1 ). A bialgebra H is called a Long skew bialgebra in [10] if there exists a linear map σ : H H k, such that for any x, y, z H, (L3 ) σ(x, yz) = Σσ(x 2, y)σ(x 1, z), (L5 ) σ(xy, z) = Σσ(x, z 1 )σ(y, z 2 ), and (L1), (L2) and (L4) hold. Assume that (H, σ) is a Long bialgebra. If for any x, y H, we have (L1 ) Σσ(x, y 1 )y 2 = Σσ(x, y 2 )y 1, then we call (H, σ) a strongly Long bialgebra. Assume that (H, σ) is a strongly Long bialgebra. Then, by (L1 ) and (L5), for any h, g H, σ(hg, ) = σ(gh, ). So, by Proposition 2.1 in [10], if σ is invertible, we know that (H, σ) is a Yang-Baxter coalgebra in [11]. In what follows, we give some examples of strongly Long bialgebras.

3 6Ï Úa, ±ÿ,],üû: The Maschke Theorem for L-R-smash Products 659 Example 1.1 Let H = H 4 be a Sweedler s 4-dimensional Hopf algebra for a given field k of chark 2. Then, by [7], it is described as follows: with coalgebra structure H 4 = k 1, x, y, xy x 2 = 1, y 2 = 0, yx = xy (x) = x x, (y) = y 1 + x y, ε(x) = 1, ε(y) = 0 whose antipode is given by S(x) = x 1, S(y) = xy. Let σ : H H k be a k-linear map as follows: σ(x, 1) = 1 = σ(1, x) = σ(x, x), σ(y, ) = 0 = σ(, y) σ(xy, ) = 0 = σ(, xy). Then, by [9], it is easy to show that (H, σ) is a strongly Long bialgebra. Example 1.2 Let H = k x i i = 1, 2,, 6 be a free algebra generated by six generators. Its comultiplication and the counity ε are given by (x 1 ) = x 1 x 1, (x 2 ) = x 2 x 2, (x 3 ) = x 3 x 2 + x 4 x 3 + (x 2 x 3 x 4 ) x 5, (x 4 ) = x 4 x 4 + (x 2 x 3 x 4 ) (x 2 x 5 x 6 ), (x 5 ) = x 5 x 2 + (x 2 x 5 x 6 ) x 3 + x 6 x 5, (x 6 ) = (x 2 x 5 x 6 ) (x 2 x 3 x 4 ) + x 6 x 6, ε(x 1 ) = ε(x 2 ) = ε(x 4 ) = ε(x 6 ) = 1, ε(x 3 ) = ε(x 5 ) = 0. Now we denote c 11 = x 1, c 22 = x 2, c 32 = x 3, c 33 = x 4, c 42 = x 5, c 44 = x 6 and define the map φ : {x 1, x 2, x 3, x 4 } k by φ(x 1 ) = 1, φ(x 2 ) = φ(x 3 ) = φ(x 4 ) = 2, and σ(c iv, c ju ) = δ uv δ φ(xi)vδ φ(xj)v. Then, by [9], (H, σ) is a Long bialgebra, and satisfies the condition Σσ(x, y 1 )y 2 = Σσ(x, y 2 )y 1, for any x, y H. 2 Generalized L-R-smash Products Let H be a bialgebra, and A be an H-bimodule algebra with actions and, and B be an H-bicomodule algebra with coactions ρ B and ρ+ B. A generalized L-R-smash product A B is defined on a tensor product A B whose multiplication is given by (a b)(c d) = Σ(a d 1 )(b [ 1] c) b [0] d 0, for any a, c A, b, d B, where ρ B (b) = b [ 1] b[0] and ρ + B (d) = d [0] d[1]. From [5] we know that the tensor product A B with the above multiplication is an algebra with unit 1 A 1 B.

4 660 ê Æ? Ð 39ò Let A B be the generalized L-R-smash product. If A B is a bialgebra, then it is called the generalized L-R-smash product bialgebra. Proposition 2.1 Let A be an H-bimodule bialgebra, B be an H-bicomodule bialgebra, and A B be the generalized L-R-smash product. Then A B is a bialgebra if and only if for any a, c A, b, d B, (LR1) Σa 1 d 1 1 d 1 0 a 2 d 2 1 d 2 0 = Σa 1 d 1 1 d 0 1 a 2 d 1 2 d 0 2, (LR2) Σb 1[ 1] c 1 b 1[0] b 2[ 1] c 2 b 2[0] = Σb [ 1]1 c 1 b [0]1 b [ 1]2 c 2 b [0]2. The comultiplication and the counit of A B are given by (a b) = Σa 1 b 1 a 2 b 2, ε(a b) = ε A (a)ε B (b). In particular, if taking B = H then it is an H-bicomodule bialgebra whose comodule structure is given by its comultiplication H. Hence, the usual L-R-smash product A H is a bialgebra if and only if for any a A, h H, (LR1 ) Σa h 1 h 2 = Σa h 2 h 1, (LR2 ) Σh 1 a h 2 = Σh 2 a h 1. If A and H are Hopf algebras, then A H is also a Hopf algebra whose antipode is given by S(a h) = ΣS H (h 3 ) S A (a) S H (h 2 ) S H (h 1 ). Proof Suppose (LR1) and (LR2) hold. Then, for any a b, c d A B, we have (a b) (c d) = Σ(a 1 b 1 a 2 b 2 )(c 1 d 1 c 2 d 2 ) = Σ(a 1 b 1 )(c 1 d 1 ) (a 2 b 2 )(c 2 d 2 ) = Σ(a 1 d 1 1 )(b 1[ 1] c 1 ) b 1[0] d 1 0 (a 2 d 2 1 )(b 2[ 1] c 2 ) b 2[0] d 2 0 = Σ(a 1 d 1 1 )(b [ 1]1 c 1 ) b [0]1 d 0 1 (a 2 d 1 2 )(b [ 1]2 c 2 ) b [0]2 d 0 2 = (Σ(a d 1 )(b [ 1] c) b [0] d 0 ) = ((a b)(c d)). It is easy to check that the counit ε of A B is an algebra morphism, so the generalized L-R-smash product A B is a bialgebra. Conversely, if A B is a bialgebra, then, by the above proof, we know that Σ(a 1 d 1 1 )(b 1[ 1] c 1 ) b 1[0] d 1 0 (a 2 d 2 1 )(b 2[ 1] c 2 ) b 2[0] d 2 0 = Σ(a 1 d 1 1 )(b [ 1]1 c 1 ) b [0]1 d 0 1 (a 2 d 1 2 )(b [ 1]2 c 2 ) b [0]2 d 0 2. In the above equality, if taking c = 1 A and b = 1 B, then we get (LR1), and in the above equality, if taking a = 1 A and d = 1 B, then we get (LR2). If B = H, then the conditions (LR1) and (LR2) are respectively changed into (LR1 ) Σa 1 h 2 h 1 a 2 h 4 h 3 = Σa 1 h 3 h 1 a 2 h 4 h 2, (LR2 ) Σh 1 a 1 h 2 h 3 a 2 h 4 = Σh 1 a 1 h 3 h 2 a 2 h 4. It is easy to see that the above conditions (LR1 ) and (LR2 ) are respectively equivalent to the conditions (LR1 ) and (LR2 ). The left proof is straightforward.

5 6Ï Úa, ±ÿ,],üû: The Maschke Theorem for L-R-smash Products 661 Remark 2.2 In the above proposition, it is easy to show that the map φ 1 : A B A B, a d Σa d 1 d 0 is a comultiplicative map if and only if (LR1) holds, and the map φ 2 : A B A B, c b Σb [ 1] c b [0] is a comultiplicative map if and only if (LR2) holds. Let H be a Hopf algebra with bijective antipode S H, Ψ be an H-bimodule algebra, and U be an H-bicomodule algebra. A generalized diagonal crossed product in [5] Ψ U is defined on a tensor product Ψ U whose multiplication is given by (ϕ u)(ϕ u ) = ϕ(u 0 [ 1] ϕ S 1 H (u 1 )) u 0 [0] u for any ϕ, ϕ Ψ, u, u U. Then, by [5], the generalized diagonal crossed product Ψ U is an associative algebra with unit 1 Ψ 1 U. Proposition 2.3 Let H be a Hopf algebra with bijective antipode S H, Ψ be an H- bimodule bialgebra and U be an H-bicomodule bialgebra, and Ψ U denote the generalized diagonal crossed product. Then Ψ U is a bialgebra if and only if for any ϕ Ψ, u U, (LR3) Σu 1 0 [ 1] ϕ 1 S 1 H (u 1 1 ) u 1 0 [0] u 2 0 [ 1] ϕ 2 S 1 H (u 2 1 ) u 2 0 [0] = Σu 0 [ 1]1 ϕ 1 S 1 H (u 1 2) u 0 [0]1 u 0 [ 1]2 ϕ 2 S 1 H (u 1 1) u 0 [0]2. The comultiplication and the counit of Ψ U are given by (ϕ u) = ϕ 1 u 1 ϕ 2 u 2, ε(ϕ u) = ε Ψ (ϕ)ε U (u) for any ϕ Ψ, u U. In particular, if taking U = H then it is an H-bicomodule bialgebra whose comodule structure is given by its comultiplication H. Hence, the generalized diagonal crossed product Ψ H is a bialgebra if and only if for any ϕ Ψ, h H, (LR3a) Σh 1 ϕ h 2 = Σh 2 ϕ h 1, (LR3b) Σϕ S 1 H (h 1) h 2 = Σϕ S 1 H (h 2) h 1. In this case, if Ψ and H are Hopf algebras, then Ψ H is also a Hopf algebra whose antipode is given by S(ϕ h) = ΣS H (h 3 ) S Ψ (ϕ) h 2 S H (h 1 ). Proof The proof is similar to Proposition 2.1, we leave it to reader. Suppose that H is a Hopf algebra with bijective antipode S H. Then, it is easy to see that the condition (LR3b) follows from the condition (LR1 ) and the condition (LR3a) follows from the condition (LR2 ), so, by Proposition 2.1 and 2.3, if the usual L-R-smash product A H is a bialgebra(hopf algebra), then the diagonal crossed product A H is also a bialgebra (Hopf algebra). Proposition 2.4 Suppose that H is a Hopf algebra with bijective antipode S H, and the usual L-R-smash product A H is a bialgebra(hopf algebra). Then, there exists an isomorphism of bialgebras(hopf algebras): Ϝ : A H A H, a h Σa h 2 h 1

6 662 ê Æ? Ð 39ò with inverse Ϝ 1 (a h) = Σa S 1 (h 2 ) h 1. Proof By the above discussion, the diagonal crossed product A H is also a bialgebra(hopf algebra). Again by (LR1 ), it is easy to show that Ϝ is a coalgebra map, and according to Example 2.7(2) in [5], we know that Ϝ is an isomorphism of algebras with inverse Ϝ 1. 3 The Maschke Theorem for L-R-smash Products In this section, we give a Maschke Theorem for the L-R-smash product A H. Lemma 3.1 If for any h H, a A, Let H be a Hopf algebra with antipode S H, and A be an H-bimodule algebra. (LR1 ) Σa h 2 h 1 = Σa h 1 h 2, then (A, ) is a left H-module algebra, where the action is given by Proof The proof is straightforward. Lemma 3.2 h a = Σh 1 a S H (h 2 ). Let H be a Hopf algebra, A be an H-bimodule algebra, A H denote the L-R-smash product, and (A#H, ) denote the induced smash product given in Lemma 3.1. If for any h H, a A, (LR1 ) Σa h 2 h 1 = Σa h 1 h 2, then there exists an algebra isomorphism: with inverse ϕ : A#H A H, a#h Σa h 2 h 1, ψ : A H A#H, a h Σa S 1 H (h 2) h 1. Proof By the direct computation, we have ψϕ = id and ϕψ = id. Also, it is not difficult to check that the map ϕ is an algebra map. According to the above lemma and the Maschke theorem in [12], we obtain Theorem 3.3(Maschke Theorem) Let H be a finite dimensional semisimple Hopf algebra, A be an H-bimodule algebra satisfying the condition (LR1 ), and A H denote the L-R-smash product. If A is semisimple, then A H is also semisimple. Let (H, σ) be a Long skew bialgebra. Define two actions on H: x h = Σσ(x, h 2 )h 1 ; h x = Σσ(h 2, x)h 1. Then, it is not difficult to check that (H, σ,, ) is an H-bimodule algebra, and hence we have the L-R-smash product H H whose multiplication is given by (h x)(g y) = Σσ(h 2, y 2 )σ(x 1, g 2 )h 1 g 1 x 2 y 1. If (H, σ) is a strongly Long bialgebra, then it is easy to see that (H, σ) is a Long skew bialgebra such that condition (LR1 ) holds. So, by Theorem 3.3, we have

7 6Ï Úa, ±ÿ,],üû: The Maschke Theorem for L-R-smash Products 663 Corollary 3.4 Let (H, σ) be a strongly Long bialgebra and semisimple. Then the L-Rsmash product H H is semisimple. Let (H, σ) be a braided Hopf algebra in [7]. Define two actions on H: x h = Σσ(x, h 1 )h 2, h x = Σσ(h 2, S H (x))h 1. Then, for any x, y, h H, (x h) y = Σσ(x, h 1 )σ(h 3, S H (y))h 2 = x (h y). It is not difficult to prove that (H, σ,, ) is both a left H-module algebra via and a right H-module algebra via, then (H, σ,, ) is an H-bimodule algebra. Hence, we have the L-R-smash product H H whose multiplication is given by (h x)(g y) = Σσ(h 2, S H (y 2 ))σ(x 1, g 1 )h 1 g 2 x 2 y 1. So, by Theorem 3.3, if H is a finite dimensional cocommutative semisimple braided Hopf algebra, then the L-R-smash product H H is also semisimple. Hence, we obtain Corollary 3.5 Let H be a finite dimensional cocommutative semisimple braided Hopf algebra. Then the L-R-smash product H H is semisimple. According to Proposition 1 in [6] and Theorem 3.3, we get the Maschke theorem for twisted smash products. Corollary 3.6 Let H be a finite dimensional cocommutative semisimple Hopf algebra, A be an H-bimodule algebra, and A H denote the twisted smash product. If A is semisimple, then A H is semisimple. Remark 3.7 Let k be a field of characteristic zero and H be a finite dimensional cocommutative Hopf algebra. Then the Drinfel d double D(H) is semisimple. In fact, the cocommutativity of H implies S 2 = id, so, the twisted smash product H H = D(H). By Larson-Radford theorem(see Theorem in [13]), H is a finite dimensional semisimple and cosemisimple Hopf algebra, so, by Corollary in [7], D(H) is semisimple. In particular, if H = U(g) is an enveloping algebra of a Lie algebra g, then H is a cocommutative Hopf algebra and hence D(H) is semisimple. According to Proposition 2.4 and Theorem 3.3, we obtain the Maschke theorem for diagonal crossed products. Corollary 3.8 Let H be a finite dimensional semisimple Hopf algebra, A be an H-bimodule algebra, and A H denote the diagonal crossed product. If (LR1 ) holds and A is semisimple, then A H is semisimple. References [1] Bieliavsky, P., Bonneau, P., Maeda, Y., Universal deformation formulae, symplectic Lie groups and symmetric spaces, Pacific J. Math., 2007, 23: [2] Bieliavsky, P., Bonneau, P., Maeda, Y., Universal deformation formulae for three-dimensional solvable Lie groups, in: Quantum Field Theory and Noncommutative Geometry, Lecture Notes in Phys., Berlin: Springer, 2005,

8 664 ê Æ? Ð 39ò [3] Bonneau, P., Gerstenhaber, M., Giaquinto, A., Sternheimer, D., Quantum groups and deformation quantization: Explicit approaches and implicit aspects, J. Math. Phys., 2004, 45: [4] Bonneau, P., Sternheimer, D., Topological Hopf algebras, quantum groups and deformation quantization, in: Hopf Algebras in Noncommutative Geometry and Physics, Lecture Notes in Pure and Appl. Math., New York: Marcel Dekker, 2005, [5] Panaite, F., Oystaeyen, F.V., L-R-smash product for (quasi-)hopf algebras, J. Algebra, 2007, 309: [6] Zhang Liangyun, L-R smash products for bimodule algebras, Progress in Natural Science (English Series), 2006, 16(6): [7] Montgomery, S., Hopf algebras and Their Actions on Rings, CBMS Vol.82, Chicago: AMS, [8] Wang Shuanhong, Li Jinqi, On twisted smash product for bimodule algebras and the Drinfel d double, Comm. Algebra, 1998, 26(8): [9] Militaru, G., A class of non-symmetric solutions for the integrability condition of the Knizhnik-Zamolodchikov equation: a Hopf algebra approach, Comm.Algebra, 1999, 27(5): [10] Zhang Liangyun, Long bialgebras, bimodule algebras and quantum Yang-Baxter modules over Long bialgebras, Acta Mathematica Sinica, 2006, 22(4): [11] Kauffman, L., Radford, D.E., Quantum algebras, quantum coalgbras, invariants of 1-1 tangles and knots, Comm. Algebra, 2000, 28(11): [12] Cohen, M., Fishman, D., Hopf algebra actions, J. Algebra, 1986, 100: [13] Dăscălescu, S., Năstăsescu, C., Raianu Ş., Hopf algebras: An Introduction, New York: Marcel Dekker, 2001, 'ul-r-smashèmaschke½n Úa, ±ÿ, ], Üû (H à ŒÆnÆ, H, ô ) Á : Ñ2ÂL-R-smashȽ2ÂézÈ V ê(hopf ê) 7 ^, Ì ÑL-R-smashÈMaschke½n,u ÛsmashÈÚézÈ Maschke½n. ' c: V ê; V{ ê; L-R-smashÈ; Û smashè; ézè