INVARIANCE UNDER TWISTING

Transcription

1 INVINCE UNDE TWISTING Pascual Jara 1, Javier López Peña 1, Florin Panaite 2 and Freddy Van Oystaeyen 3 1 Department of lgebra, University of Granada, vda. Fuentenueva s/n, E Granada, Spain s: pjara@ugr.es, jlopez@ugr.es 2 Institute of Mathematics of the omanian cademy, PO-ox 1-764, O ucharest (omania) Florin.Panaite@imar.ro 3 Department of Mathematics and Computer Sciences, University of ntwerp, Middelheimlaan 1, ntwerp (elgium) fred.vanoystaeyen@ua.ac.be bstract We establish the so-called Invariance under Twisting Theorems, stating that twisted tensor products built up from different algebras that relate in certain ways are canonically isomorphic. These results generalize (and provide categorical versions of) several well-known results in Hopf algebra theory, such as the invariance of the smash product under Drinfeld twisting, or the isomorphism between the Drinfeld double and a certain smash product for quasitriangular Hopf algebras. INTODUCTION The theory of algebra factorizations, also known as twisted tensor products, shows up in different areas of mathematics, in strong relations with geometrical properties, number theory, and Hopf algebra theory. Since their categorical origins under the name of distributive laws (cf. [2, 28, 25]) and their more structural version given in [20, 29], many papers dealing with these structures, different interpretations of them, and their applications have appeared (cf. for instance [19, 23, 9, 30, 3, 4, 7, 10, 14, 18, 17, 16], and references therein). Precisely in Hopf algebra theory we can find a big number of interesting examples illustrating the theory of algebra factorizations, and thus it is a valuable field to be taken into account when studying any properties related to these factorizations, both because of its richness in examples and used as a helpful source of inspiration. In the present work, we want to put some feedback in this relationship, and use some techniques arising from the study of the structural properties of twisted tensor products in order to recover and unify several well-known results in the realm of Hopf algebras. It is rather frequent to find in the literature various results stating that two given objects, constructed in different ways, are isomorphic. mong these, many of them use some additional properties of the objects of study that are really different in their nature, in spite of how similar the final results might look. For instance, it is possible to find results stating the invariance of a smash product under the deformation given by a cocycle twist (cf. [24]), results stating that well-known objects from the theory of quantum groups are isomorphic to simpler ones, as it is the case for the Drinfeld double of a quasitriangular finite dimensional Hopf algebra (cf. [21]), 85

2 results in which a nontrivial braiding of the tensor product, or a nontrivial smash product, yield a trivial deformation (cf. [12, 13]). Despite the fact that these results look different and are proved under very different assumptions, it is not hard to see that they seem to share some common structural points. The aim of the present work is to outline what exactly these common points are, and how can they be used to recover all these results in a unified, more general way. This will be achieved through our Invariance Theorems in two steps, the first one dealing with a particular kind of (inner) deformation of algebras, obtained as a generalization of Ferrer Santos and Torrecillas twisting data (cf. [11]), and in a second one without putting any condition on how the deformed product is build. In the first section we will recall the above mentioned results that stand as our main motivation, as well as some basic facts concerning the definition of twisted tensor products. Main results (the two invariance theorems) are Theorems 2.4 and 2.10 (plus their right-handed analogues), which are stated and proved in Section 2, where we also show how all the motivating results can be recovered as particular cases of ours. Finally, in Section 3 we will give an iterated version of the invariance theorem, following the spirit of [14]. s an application of the iterated version of the invariance theorem, we recover some results from [5]. The present work is mostly based on Section 4 of [14], though most of the material in there has been rearranged for the sake of readability. lso, proofs of the main theorems, that in [14] were given using a Sweedler-type notation for algebras in the category of vector spaces over a fixed field k, have been rewritten in an element-free form, heavily relying on braiding notation, for which we refer to [22], [15] and [27]. These new proofs make the results valid for algebras in arbitrary (strict) monoidal categories. lso, one of the assumptions in the original form of Theorem 3.1 has been dropped, obtaining a more general result. It is also worth noting that, unlike some classical isomorphism theorems, our results do not require the use of techniques such as the existence of PW bases, hence allowing a wider range of applicability. ll the results stated in this note are valid for arbitrary (strict) monoidal categories. Throughout we will assume that we have fixed a (strict) monoidal category C with a tensor product. ll objects will be assumed to be objects in C, all maps will be assumed to be morphisms between objects in C. lgebras will be assumed to be associative, unital algebras in C. 1 PELIMINIES In this section, we will recall four apparently unrelated results appeared in the literature, and find some common points among them that will lead us to the statements of our invariance theorems. 1.1 THE MOTIVTION Invariance under Drinfeld twisting Let H be a bialgebra and F H H a 2-cocycle, that is, F is an invertible element of H H that satisfies (ε id)(f) = (id ε)(f) = 1, (1 F)(id )(F) = (F 1)( id)(f). 86

3 We denote F = F 1 F 2 and F 1 = G 1 G 2. We denote by H F the Drinfeld twist of H, which is a bialgebra having the same algebra structure as H and comultiplication given by F (h) = F (h)f 1, for all h H. If is a left H-module algebra (with H-action denoted by h a h a), the invariance under twisting of the smash product #H is the following result (see [24], [6]). Define a new multiplication on, by a a = (G 1 a)(g 2 a ), for all a,a, and denote by F 1 the new structure; then F 1 is a left H F -module algebra (with the same action as for ) and we have an algebra isomorphism F 1#H F #H, a#h G 1 a#g 2 h. (1) The Drinfeld double of a quasitriangular Hopf algebra Let H be a finite dimensional Hopf algebra with antipode S. We will work with the realization of the Drinfeld double on H cop H (see for instance [23]). well-known theorem of Majid (see [21]) asserts that if (H,r) is quasitriangular then the Drinfeld double of H is isomorphic to an ordinary smash product. More explicitely, for the realization of D(H) we work with, the isomorphism is given as follows. First, we have a left H-module algebra structure on H, denoted by H, given by (we denote r = r 1 r 2 ): h ϕ = h 1 ϕ S 1 (h 2 ), ϕ ϕ = (ϕ S 1 (r 1 ))(r 2 1 ϕ S 1 (r 2 2)), for all h H and ϕ,ϕ H, and then we have an algebra isomorphism Decoupling the smash product H #H D(H), ϕ#h ϕ S 1 (r 1 ) r 2 h. (2) ecall the following result of G. Fiore from [12], in a slightly modified (but equivalent) form. Let H be a Hopf algebra with antipode S and a left H-module algebra. ssume that there exists an algebra map ϕ : #H such that ϕ(a#1) = a for all a. Define the map : H H, (h) = ϕ(1#s(h 1 )) h 2. Then is an algebra map from H to #H and the smash product #H is isomorphic to the ordinary tensor product H Unbraiding the braided tensor product ecall the following result from [13], with a different notation and in a slightly modified (but equivalent) form, adapted to our purpose. Let (H,r) be a quasitriangular Hopf algebra, H + and H two Hopf subalgebras of H such that r H + H (we will denote r = r 1 r 2 = 1 2 H + H ). Let be a right H + -module algebra and C a right H -module algebra (actions are denoted by ), and consider their braided product C (see for instance [23]), which is just the twisted tensor product C, with twisting map given by : C C, (c b) = b r 1 c r 2. ssume that there exists an algebra map π : H + # (where H + # is the right-handed smash product) such that π(1#b) = b for all b. Define the map : C C, (c) = π(r 1 #1) c r 2. 87

4 Then is an algebra map from C to C and the braided tensor product C is isomorphic to the ordinary tensor product C (hence the existence of π allows to unbraid the braided tensor product; many examples where this happens may be found in [13], especially coming from quantum groups). 1.2 THE TOOLS The four results mentioned in the former section, though apparently unrelated, share some common points. Indeed, all of them have the same basic structure: Two algebras, X and Y, possibly with some extra structure, twisted tensor product Z = X Y, nother algebra structure X with the same underlying object (the vector space) as X, Yet another twisted tensor product Z = X Y, n algebra isomorphism Z Z. The purpose of this work is to find a general result that includes all the former ones, just relying on the fact that the algebras involved in the former results are all twisted tensor products. We recall first the basics about algebra factorizations and twisted tensor products. Let C be an algebra. factorization structure on C (or an algebra factorization of C) consists on,, subalgebras of C, such that the associated linear map ψ : C, ψ := m C (i i ) is a (linear) isomorphism, where i : C, i : C stand for the canonical inclusions of and into C. In this case, we shall also say that C is a twisted tensor product of and. So, having a factorization structure of C means that we can find two suitable sub-objects (the subalgebras and ) such that they generate the whole algebra C and doing it in a nonredundant way. The fact that the map ψ is an isomorphism has an immediate consequence: the algebra C (or more precisely, its underlying object) has to be isomorphic, as an object, to the algebraic tensor product. natural question arises: given and two k algebras, is there any way to describe all factorization structures on? The answer to this question, whenever and are unital algebras, is given by the following result: Theorem 1.1 ([29], [23], [9]). Twisted tensor products between and are in one-to-one correspondence with linear maps : satisfying the following conditions: ( m ) = (m ) ( ) ( ), (3) (m ) = ( m ) ( ) ( ), (4) (u ) = u, ( u ) = u. (5) It is straightforward checking that the conditions given in the former theorem are equivalent to the fact that the map m := (m m ) ( ) is an associative product on having u u as a unit. Whenever satisfies those conditions, we say that is a (unitary) twisting map, and we denote by := (,m ) the factorization structure associated to it, which throughout will be called the twisted tensor product of and associated to the twisting map 88

5 . Essentially, if we put the classical flip τ(b a) := a b in the place of, we recover the well known algebra structure of the tensor product, henceforth, we might look at the construction of a twisted tensor product as the replacement of this map τ by a suitable map that is good enough to give us an algebra structure. In the particular case we are dealing with algebras over a field, we may use a Sweedler-type notation, and denote by (b a) = a b = a r b r, for a, b, then (3) and (4) may be rewritten as: (aa ) b = a a r (b ) r, (6) a (bb ) = (a ) r b r b. (7) In braiding notation, we will represent a twisting map : by a crossing where we will omit the label when there is no risk of confusion, and equations (3) and (4) are represented respectively by, and whilst unitality conditions (5), in braiding notation, read and If 1, 2 C and 3 C are twisted tensor products of algebras, the twisting maps 1, 2, 3 are called compatible if they satisfy the braid relation: ( 2 ) ( 3 ) (C 1 ) = ( 1 C) ( 3 ) ( 2 ), (8) see [14]. If this is the case, the maps T 1 :C ( 1 ) ( 1 ) C and T 2 : ( 2 C) ( 2 C) given by T 1 := ( 2 ) ( 3 ) and T 2 := ( 1 C) ( 3 ) are also twisting maps and T2 ( 2 C) ( 1 ) T1 C; this algebra is denoted by 1 2 C. This construction may be iterated to an arbitrary number of factors, see [14] for complete details. 2 THE ESULTS The following results are based upon [14, Sec. 4], with slightly different notation. Proofs have been rewritten using braiding notation in order to extend the original results that appeared in [14] to a more general framework. In what follows, we shall assume that we work in a (strict) monoidal category, and that all maps are morphisms in the category (in order to recover the original results in [14], consider the category of vector spaces over the base field k). 89

6 Consider, two algebras in our category, and let : a map, denoted by. ssume that we are given two maps, µ : and :, denoted by µ and satisfying the following conditions: µ (u ) =, (9) u = u u, (10) m ( µ) ( u ) =, (11) which, in braiding notation, read respectively as follows: µ µ Now, let us define the map : by := m ( µ) ( ), (12) and denote it by := µ (13) Under some further assumptions, we can ensure that is an associative product, providing thus a different algebra structure on. More concretely, we have the following result: Proposition 2.1. With notation as above, if we have the further conditions µ ( ) = m ( µ) ( m ) ( ) ( ), (14) = (m m ) ( ) ( ), (15) then (,,u ) is an associative unital algebra, denoted in what follows by d. Proof. The fact that u is a unit for is immediately deduced from (9), (10), (11). In order to prove the associativity of, first realize that conditions (14) and (15) are written in braiding notation as µ µ and 90

7 respectively. Now we have µ (15) µ µ (14) µ proving associativity, as we wanted to show. emark 2.2. The datum in Proposition 2.1 is a generalization of the left-right version of a socalled left twisting datum in [11], which is obtained if is a bialgebra and the map is given by (b a) = b 1 a b 2. ealize that insofar we have put no restriction on the map. If we require it to be a twisting map, condition (15) boils down to requiring the map to be an algebra morphism from d to. s a consequence of Proposition 2.1 we can obtain the following result from [1]: Corollary 2.3. ([1]) Let H be a bialgebra and a right H-comodule algebra with comodule structure H, a a (0) a (1), together with a linear map H, h a h a, satisfying 1 a = a, h 1 = ε(h)1, for all h H, a, and (h 2 a) (0) h 1 (h 2 a) (1) = h 1 a (0) h 2 a (1), (16) h (a a ) = (h 1 a (0) )(h 2 a (1) a ), (17) where we denoted a a = a (0) (a (1) a ). Then (,,1) is an associative algebra. Proof. We take = H and : H H, (h a) = h 1 a h 2. Then (14) is exactly (17), and (15) is an easy consequence of (16) and of the fact that is a comodule algebra. The deformation defined via the datum (,,µ) allows us to recover the deformed product in F 1 defined by a cocycle twist, or the deformed product in H used in the Drinfeld double. The next theorem will show how can we relate this kind of deformations with the given isomorphisms. In order to do this, we will assume that all the hypotheses of Proposition 2.1 are satisfied. Moreover, we will require to be a twisting map, and assume that we have another map, :, denoted by, satisfying the following conditions: u = u u, (18) m = ( m ) ( ) ( ) ( ), (19) ( m ) ( ) = u, (20) ( m ) ( ) = u, (21) 91

8 which we may also write down as Theorem 2.4 (Invariance under twisting). ssume that all the hypotheses above are satisfied; then the map d : d d defined by d := ( m ) ( m ) ( ) ( ) (22) is a twisting map, and we have an algebra isomorphism : d d given by := ( m ) ( ). Proof. In braiding notation, the maps d and are written respectively as d d d d := and := d d We begin by proving the compatibility of d with the units. On the one hand we have d d d d d (5) d (20) d d d ensuring compatibility with the unit of. On the other hand, d (10) (18) d d d d d d proving compatibility with the unit of d (which is the same as the unit of ). 92

9 Let us now check the twisting conditions for d. For (3), we have d d d d (22) d d d (15) d d d d d (19) d d (asoc.) d (asoc.) d d (4) d d (22) d d d d d d d d whilst for (4) we have d d d (22) d d (4) d (asoc.) d (21) d (asoc) d d d 93

10 (asoc) d d d d d d proving that d is a twisting map. Let us prove now that the map is an algebra isomorphism. First, in order to check that is invertible, consider the map ϕ := ( m ) ( ), and let us show that it is the inverse of. Indeed, for ϕ we have d d d d (20) d d d d n analogous proof, using (21), shows that ϕ =. It is straightforward to check that preserves the unit, so we only have to prove that it is multiplicative: d d d d d (15) d d (asoc) d d (21) (21) d d d d (4) d d as we wanted to show. 94

11 We can recover some of the results given in Section 1.1 as consequences of the Invariance Theorem. Example 2.5 (Invariance under Drinfeld twisting). For this example, under the same assumptions as in Section 1.1, let us take = H. The map : H H given by (h a) = h 1 a h 2 is a twisting map, giving us = #H. Now, we may define the maps µ : H, µ(h a) := h a, : H, (a) := a (0) a (1) := G 1 a G 2, : H, (a) := a [0] a [1] := F 1 a F 2, following the notations of Proposition 2.1 and Theorem 2.4, obtaining as the associated product on the one given by a a = a (0) (a (1) a ) = (G 1 a)(g 2 a ), which is exactly the cocycle twist of the usual product of, thus defining F 1. One can check, by direct computation, that all the necessary conditions for applying Theorem 2.4 are satisfied, hence we have the twisting map d : H F 1 F 1 H, which looks as follows: d (h a) = (a (0) ) [0] (a (0) ) [1] h a (1) = (h 1 a (0) ) [0] (h 1 a (0) ) [1] h 2 a (1) = (h 1 G 1 a) [0] (h 1 G 1 a) [1] h 2 G 2 = F 1 h 1 G 1 a F 2 h 2 G 2 = h (1) a h (2), where we denoted by F (h) = h (1) h (2) the comultiplication of H F. Hence, we obtain that d d = F 1 d H = F 1#H F, and it is obvious that the isomorphism d d provided by Theorem 2.4 coincides with the one given by (1). Example 2.6 (The Drinfeld double of a quasitriangular Hopf algebra). We take = H, with its ordinary algebra structure, = H, and : H H H H, the twisting map induced by the left and right coadjoint actions: (h ϕ) := h 1 ϕ S 1 (h 3 ) h 2, so that = D(H), the Drinfeld double of H. Denoting r 1 = u 1 u 2, the inverse of the element giving the quasitriangular structure, we define the maps: µ : H H H, µ(h ϕ) := h ϕ = h 1 ϕ S 1 (h 2 ), : H H H, (ϕ) := ϕ S 1 (r 1 ) r 2, : H H H, (ϕ) := ϕ S 1 (u 1 ) u 2, that induce on H the product given by ϕ ϕ = ϕ (0) (ϕ (1) ϕ ) = (ϕ S 1 (r 1 ))(r 2 ϕ ) = (ϕ S 1 (r 1 ))(r 2 1 ϕ S 1 (r 2 2)), which is exactly the product of H. gain, a direct computation shows that all the necessary conditions for applying Theorem 2.4 are satisfied, hence we have the twisting map d : H H H H, which looks as follows: d (h ϕ) = (ϕ (0) ) [0] (ϕ (0) ) [1] h ϕ (1) = ϕ (0) S 1 (u 1 ) u 2 h ϕ (1) = (ϕ S 1 (r 1 )) S 1 (u 1 ) u 2 h r 2 = h 1 ϕ S 1 (r 1 )S 1 (h 3 )S 1 (u 1 ) u 2 h 2 r 2 = h 1 ϕ S 1 (u 1 h 3 r 1 ) u 2 h 2 r 2 95

12 = h 1 ϕ S 1 (h 2 ) h 3 = h 1 ϕ h 2 (for the sixth equality we used the fact that cop (h)r = r (h)), hence we obtain that d d = H d H = H #H, and it is obvious that the isomorphism d d provided by Theorem 2.4 coincides with the one given by (2). Proposition 2.1 and Theorem 2.4 admit right-left versions, whose proofs are similar to the leftright versions above and therefore will be omitted: Proposition 2.7. Consider, C two algebras, and maps : C C, ν : C C and : C C such that u C = u u C, (23) ν (C u ) = C, (24) m C (ν C) (u ) = C. (25) Denote by the map : C C C given by := m C (ν C) (C ). If the following conditions are satisfied: ν ( ) = m C (ν C) (C m C) (C ) (C ), (26) = (m m C ) ( C) ( ) (27) then (C,,u C ) is an algebra, which will be denoted in what follows by d C. Theorem 2.8. ssume that the hypotheses of Proposition 2.7 are satisfied, such that moreover is a twisting map. ssume also that we are given a map : C C, such that the following relations hold: u C = u u C, (28) m C = (m ) ( C) ( C) ( ), (29) Then the map d : d C d C defined as (m C) ( ) = u C, (30) (m C) ( ) = u C. (31) d := (m C) (m ) ( ) ( ) (32) is a twisting map, and we have an algebra isomorphism d d C C given by ϕ := (m C) ( ). Example 2.9 (ight smash product). particular case of Theorem 2.8 is the invariance under twisting of the right smash product from [5]. Namely, let H be a bialgebra, C a right H-module algebra (with action denoted by c h c h) and F H H a 2-cocycle. The right smash product H#C has multiplication (h#c)(h #c ) = hh 1 #(c h 2 )c. If we define a new multiplication on C, by c c = (c F 1 )(c F 2 ) and denote the new structure by F C, then F C becomes a right H F -module algebra and we have an algebra isomorphism H F # F C H#C, h#c hf 1 #c F 2, see [5]. This result may be reobtained as a consequence of Theorem 2.8, by taking = H, (c h) = h 1 c h 2, ν(c h) = c h, (c) = F 1 c F 2, (c) = G 1 c G 2, where we denoted as before F 1 = G 1 G 2. 96

13 Whilst the Invariance Theorem allows us to recover the isomorphisms for our first two motivating exmples (the Drinfeld twist and the Drinfeld double of a quasitriangular Hopf algebra), it is not enough to recover the last two. careful look at the proof of the Invariance Theorem 2.4 shows that it does not really involve the datum used to define the deformed product, but rather only the compatibility of this product with the rest of the mappings. This fact allows us to restate the Invariance Theorem in a more general form (and, of course, the same thing holds for Theorem 2.8). More concretely, we have the following results: Theorem 2.10 (Second Invariance Theorem). Let be a twisted tensor product of algebras, consider another algebra structure on the underlying object such that u = u (that is, has the same unit as ). ssume that we are given an algebra map :, and a map :, such that the relations (18), (19), (20) and (21) are satisfied. Then the map : defined by := ( m ) ( m ) ( ) ( ) (33) is a twisting map, and we have an algebra isomorphism : given by := ( m ) ( ). Theorem Let C be a twisted tensor product of algebras, consider (C,) another algebra structure on C with u C = u C. ssume that we are given an algebra map : C C, and a map : C C, such that the relations (28), (29), (30) and (31) are satisfied. Then the map : C C defined by := (m C) (m ) ( ) ( ) (34) is a twisting map, and we have an algebra isomorphism ϕ = (m C) ( ) : C C. These extended Invariance Theorems are general enough to include our last two examples: Example 2.12 (Decoupling the smash product). We prove that Fiore s result can be recovered as a particular case of Theorem 2.11, where we take = and C = C = H (in the notation of Theorem 2.11). Define the map : H H, (h) = ϕ(1#h 1 ) h 2, and denote (h) = h < 1> h <0> and (h) = h { 1} h {0}. The relations (30) and (31) are easy to check, so we only have to prove (29) (here, the map : H H is given by (h a) = h 1 a h 2 ). We will need the following relation from [12]: for all h H, a. Now we compute: ϕ(1#h)a = (h 1 a)ϕ(1#h 2 ), (35) (h { 1} ) (h ) { 1} (h ) {0} h {0} = ϕ(1#h 1) ϕ(1#(h ) 1 ) (h ) 2 h 2 = (h 1 ϕ(1#h 1))ϕ(1#h 2 ) h 3 h 2 = ϕ(1#h 1 h 1) h 2 h 2 = (hh ), (35) = ϕ(1#h 1 )ϕ(1#h 1) h 2 h 2 hence (29) holds. pplying Theorem 2.11, we obtain the twisting map : (h a) = h < 1> a (h <0> ) { 1} (h <0> ) {0} = ϕ(1#s(h 1 ))a (h 2 ) { 1} (h 2 ) {0} 97

14 = ϕ(1#s(h 1 ))(h 2 a)(h 3 ) { 1} (h 3 ) {0} = ϕ(1#s(h 1 ))(h 2 a)ϕ(1#h 3 ) h 4 (35) = ϕ(1#s(h 1 ))ϕ(1#h 2 )a h 3 = ϕ(1#s(h 1 )h 2 )a h 3 = a h, so is the usual flip, hence we obtain #H H as a consequence of Theorem emark Let H be a Hopf algebra, let be an algebra and u : H an algebra map; consider the strongly inner action of H on afforded by u, that is, the action given by h a = u(h 1 )au(s(h 2 )), for all h H, a. Then it is well-known (see for instance [26], Example 7.3.3) that the smash product #H is isomorphic to the ordinary tensor product H. This result is actually a particular case of Fiore s theorem presented above (hence of Theorem 2.11 too), because one can easily see that the map ϕ : #H, ϕ(a#h) = au(h) is an algebra map satisfying ϕ(a#1) = a for all a. Example 2.14 (Unbraiding the braided tensor product). We prove now that the unbraiding of the braided tensor product proved by Fiore, Steinacker and Wess can also be recovered as a particular case of Theorem 2.11, where we take C = C (in the notation of Theorem 2.11). ecall first that a quasitriangular bialgebra (or Hopf algebra) is a pair (H,r), where H is a bialgebra (resp. a Hopf algebra) and r = r 1 r 2 H H such that: (QT1) (r 1 ) r 2 = r 13 r 23, (QT2) ε(r 1 )r 2 = 1, (QT3) r 1 (r 2 ) = r 13 r 12, (QT4) r 1 ε(r 2 ) = 1, (QT5) cop (h)r = r (h) for all h H, where r 12, r 13 and r 23 are the elements of H H H given by r 12 := r 1 r 2 H, r 13 = r 1 H r 2 and r 23 = H r 1 r 2, see [26] or [8] for more details. Define the map : C C, (c) = π(u 1 #1) c u 2, where we denote r 1 = u 1 u 2 = U 1 U 2 H + H. Denote as above (c) = c < 1> c <0> and (c) = c { 1} c {0}. The relations (30) and (31) are easy to check, hence we only have to prove (29) (here, we recall, coincides with the multiplication of C). We first record the relation: which can be proved as follows: c { 1} b c {0} = b (c ) { 1} (c ) {0}, b, c C, (36) b (c ) { 1} (c ) {0} = (b r 1 )(c r 2 ) { 1} (c r 2 ) {0} = (b r 1 )π(u 1 #1) c r 2 u 2 = π(1#b r 1 )π(u 1 #1) c r 2 u 2 Now we compute: = π((1#b r 1 )(u 1 #1)) c r 2 u 2 = π(u 1 1#b r 1 u 1 2) c r 2 u 2 (QT1) = π(u 1 #b r 1 u 1 ) c r 2 u 2 U 2 = π(u 1 #b) c U 2 = π(u 1 #1)π(1#b) c U 2 = π(u 1 #1)b c U 2 = c { 1} b c {0}. (cc ) = π(u 1 #1) (c u 2 1)(c u 2 2) (QT3) = π(u 1 U 1 #1) (c u 2 )(c U 2 ) = π(u 1 #1)π(U 1 #1) (c u 2 )(c U 2 ) = c { 1} c { 1} c {0}c {0} (36) = c { 1} (c ) { 1} (c ) {0} c {0}, 98

15 hence (29) holds. Theorem 2.11 may thus be applied, and we get the twisting map, which looks as follows: (c b) = c < 1> b (c <0> ) { 1} (c <0> ) {0} = π(r 1 #1)b ((c r 2 ) ) { 1} ((c r 2 ) ) {0} = π(r 1 #1)(b 1 )(c r 2 2 ) { 1} (c r 2 2 ) {0} = π(r 1 #1)π(1#b 1 )π(u 1 #1) c r 2 2 u 2 = π((r 1 #1)(1#b 1 )(u 1 #1)) c r 2 2 u 2 = π(r 1 u 1 1#b 1 u 1 2) c r 2 2 u 2 (QT1) = π(r 1 U 1 #b 1 u 1 ) c r 2 2 u 2 U 2 = π(1#b) c = b c, so is again the usual flip, hence we obtain C C as a consequence of Theorem ITETED VESION natural question that arises is to see whether Theorems 2.10 and 2.11 can be combined, namely, if (,,C, 1, 2, 3 ) are three algebras and compatible twisting maps as defined in (8) and we have a datum as in Theorem 2.10 between and and a datum as in Theorem 2.11 between and C, under what conditions does it follow that we can build an iterated twisted tensor product using (,,C, 1, 2, 3 ), that is, when can we ensure that the maps 1, 2 and 3 are compatible, for a suitable twisting map : C C. Our first remark is that this does not happen in general, since a counterexample may be obtained as follows. Take = H a bialgebra, a left H-module algebra, C a right H-module algebra and F H H a 2-cocycle. Here 1 (h a) = h 1 a h 2, 2 (c h) = h 1 c h 2 and 3 = τ C, the usual flip, hence 1 H 2 C = #H#C, the two-sided smash product. We consider the datum between and H that allows us to define F 1#H F, hence d 1 (h a) = F1 h 1 G 1 a F 2 h 2 G 2, and the trivial datum between H and C. One can see that in general ( d 1, 2, 3 ) do not satisfy the braid condition. Hence, the best we can do is to find sufficient conditions on the initial data ensuring that ( 1, 2, 3 ) satisfy the braid condition. This is achieved in the next result, which generalizes Theorem 4.10 from [14]. Theorem 3.1. Let ( 1, 2, 3 ) be compatible twisting maps providing an iterated twisted tensor product 1 2 C. ssume that we have deformation data (m,,) between and as in Theorem 2.10 and (m C,,) between and C as in Theorem 2.11, where m and m C represent the multiplications in the deformed algebras and C, and let 3 : C C be a twisting map between the deformed algebras. ssume also that the following compatibility conditions hold: ( m C) ( ) ( 2 ) ( 3 ) (C ) = ( m C) ( 1 C) ( C) ( 3) ( ), (37) ( m C) ( 2 ) ( ) ( 3 ) (C ) = ( m C) ( C) ( 1 C) ( 3 ) ( ), (38) ( m C) ( 1 2 ) ( 3 ) ( ) = ( m C) ( ) 3. (39) Then ( 1, 2, 3 ) are compatible twisting maps providing an iterated twisted tensor product 1 2 C and we have an algebra isomorphism ψ : 1 2 C 1 2 C given 99

16 by ψ := ( m C) ( m C) ( ). (40) Proof. First, in braiding notation the conditions (37), (38), and (39) are written as: C 3 C 3 C 3 C 3 C C 3 2 1, 2 1 and C C C C C C Now, we check the braid equation (8) for the maps 1, 2, 3 : C 2 C 2 C 2 C 2 3 (33),(34) 3 1 (4) 3 1 (asoc) 3 1 (37) 1 C C C C C C C 1 3 (37) 1 (8) 2 (38) (3) C C 100 C

17 C 1 (3) 3 2 C (33),(34) C C as we wanted to show. We prove now that the map ψ is an algebra isomorphism. First, using (20), (21), (30), (31), it is easy to see that ψ is invertible, its inverse being the map η := ( m C) ( m C) ( ). We prove now that ψ is multiplicative: C C C C 3 C C 3 C (33),(34) (asoc.) C C C 3 C C C 3 (asoc.) 1 2 (21),(31) C C 101

18 C C 3 C C (3),(4) 1 2 (39) C C C C 3 C C (39) (3),(4) C C C C 3 (3),(4) 1 2 C concluding the proof. Example 3.2. Let H be a bialgebra, a left H-module algebra, C a right H-module algebra and F H H a 2-cocycle. Then, by [5], we have an algebra isomorphism between the two-sided smash products: F 1#H F # F C #H#C, a#h#c G 1 a#g 2 hf 1 #c F 2. One can easily see that this result is a particular case of Theorem 3.1; indeed, the relations (37), (38), (39) are easy consequences of the 2-cocycle condition for F. CKNOWLEDGMENT Pascual Jara and Javier López have been partially supported by projects MTM and FQM-266 (Junta de ndaluca esearch Group). Javier López has also been supported by the 102

19 Spanish MEC FPU-grant P Florin Panaite and Fred Van Oystaeyen have been partially supported by the EC programme LIEGITS, TN 2003, , and by the bilateral project WS04/04 New Techniques in Hopf algebras and graded ring theory of the Flemish and omanian governments. Florin Panaite was partially supported by the programme CEEX of the omanian Ministry of Education and esearch, contract nr. 2-CEx /2006. EFEENCES [1] M. eattie, C.-Y. Chen, and J. J. Zhang, Twisted Hopf comodule algebras, Comm. lgebra 24 (1996), [2] J. eck, Distributive laws, Lect. Notes Math. 80 (1969), [3] T. rzeziński and S. Majid, Coalgebra bundles, Comm. Math. Phys. 191 (1998), [4] T. rzeziński and S. Majid, Quantum geometry of algebra factorisations and coalgebra bundles, Comm. Math. Phys. 213 (2000), [5] D. ulacu, F. Panaite, and F. Van Oystaeyen, Generalized diagonal crossed products and smash products for quasi-hopf algebras. pplications, Comm. Math. Phys. 266 (2006), [6] D. ulacu, F. Panaite, and F. Van Oystaeyen, Quasi-Hopf algebra actions and smash products. Comm. lgebra 28 (2000), [7] S. Caenepeel,. Ion, G. Militaru, and S. Zhu, The factorisation problem and smash biproducts of algebras and coalgebras. lgebr. epresent. Theory 3 (2000), [8] S. Caenepeel, G. Militaru and Shenglin Zhu, Frobenius and separable functors for generalized module categories and nonlinear equations, Lect. Notes Math. 1787, Springer Verlag, erlin, [9]. Cap, H. Schichl, and J. Vanžura, On twisted tensor products of algebras, Comm. lgebra 23 (1995), [10] C. Cibils, Non-commutative duplicates of finite sets, J. lgebra ppl. 5 (2006), [11] W.. Ferrer Santos and. Torrecillas, Twisting products in algebras II, K-Theory 17 (1999), [12] G. Fiore, On the decoupling of the homogeneous and inhomogeneous parts in inhomogeneous quantum groups, J. Phys. 35 (2002), [13] G. Fiore, H. Steinacker, and J. Wess, Unbraiding the braided tensor product, J. Math. Phys. 44 (2003), [14] P. Jara Martínez, J. López Peña, F. Panaite, and F. Van Oystaeyen, On iterated twisted tensor products of algebras, preprint arxiv:math.q/ [15] C. Kassel, Quantum Groups, Grad. Texts Math. 155, Springer Verlag, erlin, [16] J. López Peña, Connections over twisted tensor products of algebras, preprint

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