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The Factorization Problem and the Smash Biproduct of Algebras and Coalgebras

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The Factorization Problem and the Smash Biproduct of Algebras and Coalgebras Algebras and Representation Theory 3: 19–42, 2000. 19 © 2000 Kluwer Academic Publishers. Printed in the Netherlands. The Factorization Problem and the Smash Biproduct of Algebras and Coalgebras S. CAENEPEEL1, BOGDAN ION2, G. MILITARU3;? and SHENGLIN ZHU4;?? 1University of Brussels, VUB, Faculty of Applied Sciences, Pleinlaan 2, B-1050 Brussels, Belgium 2Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, U.S.A. 3University of Bucharest, Faculty of Mathematics, Str. Academiei 14, RO-70109 Bucharest 1, Romania 4Institute of Mathematics, Fudan University, Shanghai 200433, China (Received: July 1998) Presented by A. Verschoren Abstract. We consider the factorization problem for bialgebras. Let L and H be algebras and coalgebras (but not necessarily bialgebras) and consider two maps R: H ⊗ L ! L ⊗ H and W: L ⊗ H ! H ⊗ L. We introduce a product K D L W FG R H and we give necessary and sufficient conditions for K to be a bialgebra. Our construction generalizes products introduced by Majid and Radford. Also, some of the pointed Hopf algebras that were recently constructed by Beattie, Dascˇ alescuˇ and Grünenfelder appear as special cases. Mathematics Subject Classification (2000): 16W30. Key words: Hopf algebra, smash product, factorization structure. Introduction The factorization problem for a ‘structure’ (group, algebra, coalgebra, bialgebra) can be roughly stated as follows: under which conditions can an object X be written as a product of two subobjects A and B which have minimal intersection (for example A \ B Df1Xg in the group case). A related problem is that of the construction of a new object (let us denote it by AB) out of the objects A and B.In the constructions of this type existing in the literature ([13, 20, 27]), the object AB factorizes into A and B. This is the case – to give an example – with Majid’s double crossed product of Hopf algebras. Moreover, whenever a Hopf algebra factorizes in a natural way into two sub-Hopf algebras, it is likely to be a double crossed product ([15, Thm. 7.2.3]). This examples include the quantum double of Drinfel’d, and lead also to natural generalizations of it on a pairing or skew-pairing of bialgebras. ? Research supported by the bilateral project ‘Hopf algebras and co-Galois theory’ of the Flemish and Romanian governments. ?? Research supported by the ‘FWO Flanders research network WO.011.96N’. 20 S. CAENEPEEL ET AL. The simplest example of algebra factorization is the tensor product of two k- algebras or, more general, the tensor product of two algebras A and B in a braided monoidal category. The multiplication is then given by the formula mA#B D .mA ⊗ mB/ ◦ .IA ⊗ RA;B ⊗ IB/; (1) where R is the braiding on the category. In order to apply (1) to two particular algebras, A and B, we do not need a braiding on the whole category, in fact it suffices to have a map R: B ⊗ A ! A ⊗ B. This new algebra A#RB will be called a smash product, if it is associative with unit 1A#1B . We will work in the category of vector spaces over a field k and give necessary and sufficient conditions for R to define a smash product. The smash product can be determined completely by a universal property and it will also turn out that several variations of the smash product that have appeared earlier in the literature are special cases. The construction can be dualized, leading to the definition of the smash co- product of two coalgebras C and D (Section 3). The main result of this note (Section 4) is the fact that we can combine the two constructions and this leads to the definition of the smash biproduct of two vector spaces H and L that are immediately algebras and coalgebras (but not necessarily bialgebras). The smash biproduct can be characterized using coalgebra projections onto H and L and alge- bra embeddings of H and L (Theorem 4.3). Applying this result, one can see that earlier constructions due to Majid [13] and Radford [20] are in fact smash biprod- ucts. Furthermore, some of the pointed Hopf algebras that Beattie, Dascˇ alescuˇ and Grünenfelder [2] constructed using iterated Ore extensions (including classical examples like Sweedler’s four-dimensional Hopf algebra) can be viewed as smash biproducts (Theorem 5.1). As we indicated at the beginning of this introduction, the smash biproduct can be defined in an arbitrary monoidal category. Bernhard Drabant kindly informed us that this general biproduct was recently introduced by Bespalov and Drabant in [3], where it is called a cross-product bialgebra. 1. Preliminaries Let k be a field. For two vector spaces V and W,andak-linear map R: V ⊗ W ! W ⊗ V , we write X R.v ⊗ w/ D Rw ⊗ Rv for all v 2 V , w 2 W. Using this notation, the k-linear map .R ⊗ IV /.IV ⊗ R/: V ⊗ V ⊗ W ! W ⊗ V ⊗ V can be denoted as follows (we write R D r): X r R r R .R ⊗ IV /.IV ⊗ R/.v1 ⊗ v2 ⊗ w/ D . w/ ⊗ v1 ⊗ v2 SMASH BIPRODUCT OF ALGEBRAS AND COALGEBRAS 21 for all v1;v2 2 V , w 2 W. Let A be a k-algebra. mA: A ⊗ A ! A will be the multiplication map on A,and1A the unit of A.Forak-coalgebra C, 1C : C ! C ⊗ C will be the comultiplication, and "C : C ! k the augmentation map. 2. A Smash Product between Two Algebras Let A and B be associative k-algebras with unit, and consider a k-linear map R: B ⊗ A ! A ⊗ B. By definition A#RB is equal to A ⊗ B as a k-vector space, with multiplication given by the formula D ⊗ ⊗ ⊗ mA#RB .mA mB/.IA R IB/ (2) or X R R .a#Rb/.c#Rd/ D a c#R bd (3) for all a;c 2 A, b; d 2 B. This multiplication appeared first in [29], where a sufficient condition is given for mA#RB to be associative. DEFINITION 2.1. Let A and B be k-algebras with unit, and R: B ⊗ A ! A ⊗ B a k-linear map. If A#RB is an associative k-algebra with unit 1A#1B , then we call A#RB a smash product. Remark 2.2. Let R be a braiding on the category Mk of k-vector spaces. Then we have maps RX;Y : Y ⊗X ! X⊗Y , for all vector spaces X and Y , and our smash ⊗ product A#RA;B B is the usual product in the braided category .Mk; ;k;RX;Y / (see [14]). Our definition has a local character. In fact, given two algebras A and B, one can sometimes compute explicitely all the maps R that make A#RB into a smash product (see Example 2.11(3)). EXAMPLES 2.3. (1) Let R D τB;A: B ⊗ A ! A ⊗ B be the switch map. Then A#RB D A ⊗ B is the usual tensor product of A and B. (2) Let G be a group acting on the k-algebra A. This means that we have a group g homomorphism σ : G ! Autk.A/. Writing σ .g/.a/ D a,wefindak-linear map R: kG ⊗ A ! A ⊗ kG; R.g ⊗ a/ D ga ⊗ g and A#RkTGUDA ∗σ G is the usual skew group algebra. (3) More generally, let H be a Hopf algebra, A aleftH -module algebra and D aleftH -comodule algebra. Let X R: D ⊗ A ! A ⊗ D; R.d ⊗ a/ D dh−1i · a ⊗ dh0i: Then A#RD D A#D is Takeuchi’s smash product [28]. For D D H , we obtain the usual smash product A#H defined in Sweedler’s book [24]. 22 S. CAENEPEEL ET AL. (4) Let .G; H / be a matched pair of groups and H FG G the product associated to this pair (see [27]). Write G × H ! H.g;h/7! g · h; G × H ! G.g;h/7! gh for the respective group actions, and define R: kG ⊗ kH ! kH ⊗ kG; R.g ⊗ h/ D g · h ⊗ gh for all g 2 G and h 2 H .ThenkH#RkG D kTH FG GU. In Example 2.11(4), we will construct an example of a smash product kH#RkG which is not of the form kTH FG GU. DEFINITION 2.4. Let A and B be k-algebras and R: B ⊗ A ! B ⊗ A a k-linear map. R is called left normal if .LN/R.b⊗ 1A/ D 1A ⊗ b for all b 2 B. R is called right normal if .RN/R.1B ⊗ a/ D a ⊗ 1B for all a 2 B. We call R normal if R is left and right normal. In the following theorem, we give necessary and sufficient conditions for A#RB to be a smash product. THEOREM 2.5. Let A; B be two algebras and let R: B ⊗ A ! A ⊗ B be a k-linear map. The following statements are equivalent (1) A#RB is a smash product. (2) The following conditions hold: (N) R is normal; (O) the following octogonal diagram is commutative I ⊗m ⊗I R⊗I B ⊗ A ⊗ A ⊗ B B A B B ⊗ A ⊗ B B A ⊗ B ⊗ B IB ⊗IA⊗R IA⊗mB B ⊗ A ⊗ B ⊗ A A ⊗ B R⊗IB ⊗IA mA⊗IB I ⊗m ⊗I I ⊗R A ⊗ B ⊗ B ⊗ A A B A A ⊗ B ⊗ A A A ⊗ A ⊗ B (3) The following conditions hold: (N) R is normal; (P) the following two pentagonal diagrams are commutative: SMASH BIPRODUCT OF ALGEBRAS AND COALGEBRAS 23 (P1) m ⊗I B ⊗ B ⊗ A B A B ⊗ A R A ⊗ B IB ⊗R IA⊗mB R⊗I B ⊗ A ⊗ B B A ⊗ B ⊗ B (P2) I ⊗m B ⊗ A ⊗ A B A B ⊗ A R A ⊗ B R⊗IA mA⊗IB I ⊗R A ⊗ B ⊗ A A A ⊗ A ⊗ B Proof.
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