Acta Mathematica Sinica, English Series April, 2004, Vol.20, No.2, pp. 305–318
On Quasi-Bicrossed Products of Weak Hopf Algebras Fang LI Department of Mathematics, Zhejiang University, Hangzhou 310028, P. R. China E-mail:
[email protected]
Abstract The aim of this paper is to study quasi-bicrossed products and a especially quantum quasi-doubles. Firstly, we construct one new kind of quasi-bicrossed products by weak Hopf algebras and then devote a brief discussion to this matter. And, we discuss the conditions for quasi-bicrossed products to possess the structure of almost weak Hopf algebras, containing the case of a special smash product. At the end, we give some properties on the quantum quasi-double, respectively on the quasiR-isomorphism, the representation-theoretic interpretation and the regularity of the quasi-R-matrix. Keywords Weak Hopf algebra, Bicrossed product, Quantum double, R-matrix MR(2000) Subject Classification 16W30, 16W35
1
Introduction
As is known [1], a bicrossed product is a fundamental tool for constructing the quantum double of a Hopf algebra; in [2], a quasi-bicrossed product is in a similar role for the quantum quasidouble of a certain weak Hopf algebra (in particular, a finite Clifford monoid). So, in order to research more on a (quasi-)braided (almost) bialgebra, one needs to construct new (quasi-) bicrossed products. In Section 2, we introduce the concept of a weak Hopf pair as a generalization of a Hopf pair by Takeuchi [3]. Then, using weak Hopf pairs, we construct one new kind of quasi-bicrossed products, which lies between a general quasi-bicrossed product and a quantum quasi-double. Moreover, we discuss their commutativity and co-commutativity. In Section 3, we discuss the question of whether a quasi-bicrossed product possesses an almost weak antipode. It is mentioned that in general, this question is false from an example on the quantum quasi-double of a Clifford monoid. On the other hand, some examples are given as a partial positive answer to the question. In particular, for a co-commutative bialgebra L and a left K-module-bialgebra L, we point out that their smash product L#K is a special bicrossed product and verify that the smash product L#K is a weak Hopf algebra when L is a weak Hopf algebra and K is a Hopf algebra. The quasi-bicrossed product D(X, A) in Theorem 2.1 is just a quantum quasi-double in the case of X or A being finite dimensional. In Section 4 we give some properties on the Received September 11, 2000, Revised June 25, 2002, Accepted December 13, 2002 Project supported by the National Natural Science Foundation of China (No. 19971074) and also by the Natural Science Foundation of Zhejiang Province (No. 102028)
Li F.
306
quantum quasi-double, respectively on the quasi-R-isomorphism, the representation-theoretic interpretation and the regularity of the quasi-R-matrix. Now, we introduce some useful concepts. Other concepts unexplained here can be found in [4] and [1]. Here, any algebra is over a field k. A bialgebra H = (H, µ, η, ∆, ε) is called a right (resp. left) Hopf algebra if there exists S ∈ Homk (H, H) such that id ∗ S = ηε (resp. S ∗ id = ηε) and then S is called a right (resp. left) antipode of H. This concept was introduced and studied early in [5]. In [2] [6], we introduced the concept of a weak Hopf algebra as a generalization of a right (resp. left) Hopf algebra and studied its characterizations and applications. A k-bialgebra H = (H, µ, η, ∆, ε) is called a weak Hopf algebra if there exists T ∈ Homk (H, H) such that id ∗ T ∗ id = id and T ∗ id ∗ T = T , where T is called a weak antipode of H. A weak (resp. right/left) Hopf algebra H is called a perfect weak (resp. right/left) Hopf algebra if its weak (resp. right/left) antipode T is an anti-bialgebra morphism and (T ∗id)(H) ⊆ C(H) (the centre of H). We say in [2] that a linear space H is a k-almost bialgebra if (H, µ, η) is a k-algebra and (H, ∆, ε) is a k-co-algebra with ∆(xy) = ∆(x)∆(y) for x, y ∈ H. If K is a subalgebra and also a sub-co-algebra of H, then K itself is an almost bialgebra, called a sub-almost bialgebra of H. Combining formally with the definition of a weak Hopf algebra, we say in [6] that an almost bialgebra H is an almost weak Hopf algebra if there exists T ∈ Homk (H, H) such that id ∗ T ∗ id = id and T ∗ id ∗ T = T , where T is called an almost weak antipode of H. Let H be an almost bialgebra. If there exists an R ∈ H ⊗ H such that for all x ∈ H we have ∆op (x)R = R∆(x) and (∆ ⊗ id)(R) = R13 R23 , (id ⊗ ∆)(R) = R13 R12 , then we call H a quasi-braided almost bialgebra with a quasi-R-matrix R (see [2]). Moreover, if H is a bialgebra and R is invertible, then H is called a braided bialgebra. Let H be a bialgebra and C a co-algebra. If C is a left H-module and ∆(hc) = ∆(h)∆(c) for every h ∈ H and c ∈ C, then we call the co-algebra C a left quasi-module-co-algebra over H. Moreover, if ε(hc) = ε(h)ε(c), then C is called a left module-co-algebra over H. Right quasi-module-co-algebra and right module-co-algebra can be defined similarly. A pair (X, A) of bialgebras over a field k is said to be quasi-matched (resp. matched) if there exist linear maps α : A ⊗ X −→ X and β : A ⊗ X −→ A turning X into a left A-quasi-moduleco-algebra (resp. a left A-module-co-algebra) and turning A into a right X-quasi-module-coalgebra (resp. a right X-module-co-algebra) such that if one sets α(a⊗x) = ax, β(a⊗x) = ax, the following conditions are satisfied: a (xy) = (a x )((a x ) y), (1) (a)(x)
a 1 = ε(a)1, (ab) x = (a (b x ))(b x ),
(2) (3)
(b)(x)
1 x = ε(x)1, (a x ) ⊗ (a x ) = (a x ) ⊗ (a x ), (a)(x)
(a)(x)
(4) (5)
On Quasi-Bicrossed Products of Weak Hopf Algebras
307
for all a, b ∈ A and x, y ∈ X, where 1 is the identity of X or A, respectively, in (2) and (4). For a quasi-matched (resp. matched) pair of bialgebras (X, A), we know from [2] and [1] that there exists an almost bialgebra (resp. a bialgebra) structure on the vector space X ⊗ A with identity equal to 1 ⊗ 1 such that its product is given by (x ⊗ a)(y ⊗ b) = x(a y ) ⊗ (a y )b, (6) (a)(y)
its co-product by ∆(x ⊗ a) =
(x ⊗ a ) ⊗ (x ⊗ a ),
(7)
(a)(x)
and its co-unit by ε(x ⊗ a) = εX (x)εA (a),
(8)
for all x, y ∈ X, a, b ∈ A. Equipped with this almost bialgebra (resp. bialgebra) structure, X ⊗ A is called the quasi-bicrossed product (resp. bicrossed product) of X and A, and denoted as X∞A. Furthermore, the injective maps iX (x) = x ⊗ 1 and iA (a) = 1 ⊗ a from X and A, respectively, into X∞A are bialgebra morphisms. Also, x∞a = (x∞1)(1∞a) for a ∈ A and x ∈ X. 2
The Quasi-Bicrossed Product of a Weak Hopf Skew-Pair
The concept of a Hopf pair of Hopf algebras was introduced by Takeuchi in [3], which plays a valid role in studying the theory of quantum groups. Now, we generalize this concept for weak Hopf algebras. Definition 2.1 (i) Suppose that A and X are weak Hopf algebras with weak antipodes SA and SX . We call (X, A) a weak Hopf pair, if there exists a non-singular bilinear form , from X ⊗ A to k satisfying that x, ab = x , ax , b, (9) (x)
x, 1A = ε(x), xy, a = x, a y, a ,
(10) (11)
(a)
1X , a = ε(a),
(12)
SX (x), a = x, SA (a),
(13)
where x, y ∈ X, a, b ∈ A. (ii) In (i), moreover, if SA is invertible and (9) and (13) are replaced with the following (14) and (15) : x, ab = x , ax , b, (14) (x) −1 SX (x), a = x, SA (a),
(15)
we call (X, A) a weak Hopf skew-pair. −1 ) is a weak Hopf algebra when SA is invertible. ThereFrom [2], Aop = (A, µop , η, ∆, ε, SA fore (X, A) is a weak Hopf skew-pair if and only if (X, Aop ) is a weak Hopf pair in this case.
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We have known from [2] that for a finite-dimensional co-commutative perfect weak Hopf algebra H with invertible weak antipode T , the quantum quasi-double D(H) = H op∗ ∞H is a quasi-bicrossed product. We want to generalize this result to two more general perfect weak Hopf algebras. Theorem 2.2 For two perfect weak Hopf algebras A and X with weak antipodes SA and SX respectively, suppose that A is co-commutative, SA is invertible and (X, A) is a weak Hopf skew-pair. Then (X, A) is a quasi-matched pair of bialgebras with a x = (x) x SX (x ), ax −1 and a x = (a) x, SA (a )a a so as to get a quasi-bicrossed product X∞A, denoted as D(X, A). Proof
Firstly, we can easily verify the following: −1 a x, b = x, SA (a )ba ,
(16)
(a)
y, a x =
x yS(X) (x ), a,
(17)
(x)
for a, b ∈ A, x, y ∈ X. Now we prove that A and X are a right X-quasi-module co-algebra and a left A-quasimodule co-algebra, respectively, with the actions and . In fact, for any a ∈ A, x, y, z ∈ X, we have z, a (xy) = (xy) zSX ((xy) ), a = x y zSX (y )SX (x ), a = z, (a x) y, (xy)
(x)(y)
then a (xy) = (a x) y; z, a 1 = 1zSX (1), a = z, a, then a 1 = a. Thus A is a right X-module. On the other hand, y ⊗ z, (a x ) ⊗ (a x ) = y, a x z, a x (a)(x)
=
(a)(x)
x ySX (x ), a x zSX (x(4) ), a =
(a)(x)
x ySX (x )x zSX (x(4) ), a
(x)
= x yzSX (x ), a = yz, a x = y ⊗ z, ∆(a x), (x)
then ∆(ax) = (a)(x) (a x )⊗(a x ). It means that A is a right X-quasi-module-co-algebra. Similarly, we get that X is a left A-quasi-module-co-algebra. Moreover, we can see that (2) and (4) are trivial according to (12) and (10) and the definitions of and . And, −1 (4) (a x )((a x ) y), b = (a x )(x , SA (a )a a y), b (a)(x)
(a)(x)
=
−1 (4) x , SA (a )a (a x ), b (a y), b
(a)(b)(x)
=
−1 (5) −1 x , SA (a )a x , SA (a )b a (a(4) y), b
(a)(b)(x)
=
(a)(b)
−1 (5) −1 x, SA (a )a SA (a )b a (a(4) y), b
On Quasi-Bicrossed Products of Weak Hopf Algebras
=
(a)(b)
=
309
−1 x, S(A) (a )b a (a y), b −1 (4) −1 x, SA (a )b a y, SA (a )b a
(a)(b)
=
−1 xy, SA (a )ba = a (xy), b, (a)
then a (xy) = (a)(x) (a x )((a x ) y), i.e. (1) holds. Similarly, we can get that (ab) x = (b)(x) (a (b x ))(b x ), i.e. (3) holds. −1 (a x ) ⊗ (a x ) = x , SA (a )a a ⊗ x SX (x(4) ), a(4) x (a)(x)
(a)(x)
=
−1 x , SA (a )a x , a(4) SX (x(4) ), a(5) a ⊗ x
(a)(x)
=
−1 x , a(4) SA (a )a SX(x ), a(5) a ⊗ x
(a)(x)
=
−1 x , a(4) SA (a )a SX(x ), a(5) a ⊗ x
(a)(x)
=
x , a SX (x ), a a ⊗ x
(a)(x)
=
−1 x , a x , SA (a )a ⊗ x
(a)(x)
=
−1 (5) (4) −1 x , a x , SA (a )a SA (a )a ⊗ x
(a)(x)
=
−1 −1 (5) (4) x , a x , SA (a )x(4) , SA (a )a a ⊗ x
(a)(x)
=
−1 (5) (4) x , a SX (x ), a x(4) , SA (a )a a ⊗ x
(a)(x)
=
−1 (4) x SX (x ), a x(4) , SA (a )a a ⊗ x
(a)(x)
=
−1 (4) x SX (x ), a x(4) , SA (a )a a ⊗ x
(a)(x)
=
(a x ) ⊗ (a x ),
(a)(x)
then (5) holds. In a word, (X, A) is a quasi-matched pair of bialgebras. Hence we get a quasi-bicrossed product X∞A, denoted as D(X, A). We remark that in Theorem 2.1, the perfect weak Hopf algebra X must be commutative by (11) since A is co-commutative. However, certainly, Theorem 2.2 is a generalization of Theorem 2.11 in [2]. In fact, when A = H is a finite-dimensional co-commutative perfect weak Hopf algebra with invertible weak antipode, we can set X = H ∗cop and suppose , is the bilinear form of H and its duality H ∗ as linear space, then D(X, A) = D(H) the quantum quasi-double of H. In particular, for a finite Clifford monoid S, we get its quantum quasi-double D(S) = D((kS)∗cop , kS).
Li F.
310
Moreover, we can construct the following: Example 2.3 For two Clifford moniods S = {sλ : λ ∈ Λ} and T = {tν : ν ∈ Γ}, suppose that there exists a surjective homorphism π of monoids from S onto T . Let X = (kT )∗cop and A = kS where kS and kT are the semigroup Hopf algebras of S and T . Then A and X have bases {sλ : λ ∈ Λ} and {t∗ν : ν ∈ Γ} respectively, where t∗ν is the dual map of tν in Homk (kT, k). Define , by t∗ν , sλ = t∗ν (π(sλ )) and extend it bilinearly. Thus , becomes a non-singular bilinear form on X ⊗ A. It is easy to see that (10), (12), (14) and (11) are all satisfied. By Proposition 1.2 in [2], SX equals the dual map of the inverse of the antipode of kT , then −1 −1 ∗ −1 ∗ SX (t∗ν ), sλ = t∗ν SkT (π(sλ )) = t∗ν (π(sλ ))−1 = t∗ν (π(s−1 λ )) = tν , sλ = tν , SA (sλ ),
i.e. (15) holds. So, (X, A) becomes a weak Hopf pair. Then X and A are two perfect weak Hopf algebras satisfying the condition of Theorem 2.2. Hence we get a quasi-bicrossed product D(X, A). How does one find a quasi-R-matrix of D(X, A) such that D(X, A) becomes a quasi-braided almost bialgebra in the case where X and A are both infinite dimensional? Practically this is a difficult question. However, in other cases, it is trivial. In fact, since , is a non-singular bilinear form on X ⊗ A, we can view this as X ⊆ A∗ and A ⊆ X ∗ . Assume dimX < +∞ or dimA < +∞. For example, dimX < +∞. Then dimA ≤ dimX ∗ = dimX ≤ dimA∗ = dimA. It follows that X = A∗ and A = X ∗ as linear −1 spaces. From (14), (10), (11), (12) and (15), we get that X = A∗cop and SX = SA as weak Hopf algebras. Therefore, D(X, A) = D(A), the quantum quasi-double of A. Denote by GQDI the set of all quasi-bicrossed products in Theorem 2.2, by QQD the set of all quantum quasi-doubles of finite-dimensional co-commutative perfect weak Hopf algebras with invertible weak antipodes, by QBAB the set of all quasi-braided almost bialgebras. Then, QQD ≤ GQDI ∩ QBAB. The question mentioned above is equivalent to under what condition the equality holds for this including relation. Now, we give a lemma which will be necessary in Section 4. Under the condition of Theorem 2.1, for x, y ∈ X, a, b ∈ A, −1 −1 (x∞a)(y∞b) = y , a y , SA (a )xy ∞a b = xy, SA (a )?a ∞a b
Lemma 2.4
(y)(a)
(a)
where ? denotes any element of A. Proof (x∞a)(y∞b) =
x(a y )∞(a y )b
(a)(y)
=
−1 (4) xy SX (y ), a y ∞y (4) , SA (a )a a b
(a)(y)
=
−1 (5) (4) xy , a SX (y ), a y ∞y (4) , SA (a )a a b
(a)(y)
=
(a)(y)
−1 −1 (5) xy , a y y , SA (a )y (4) , SA (a )a ∞a(4) b
On Quasi-Bicrossed Products of Weak Hopf Algebras
=
(a)
=
311
−1 (5) −1 xy, SA (a )a S(A) (a )?a ∞a(4) b = −1 xy, SA (a )?a ∞a b
(a)
=
(a)
−1 (5) −1 xy, SA (a )?a S(A) (a )a ∞a(4) b
−1 y , a y , SA (a )xy ∞a b.
(y)(a)
The non-commutativity and non-co-commutativity of a quantum group is one of its pivotal properties. For this reason, let us discuss the non-commutativity and non-co-commutativity of D(X, A) in Theorem 2.2 and, in particular, quantum quasi-doubles. ∼ X∞1 and A = ∼ 1∞A are subalgebras of D(X, A) and x∞a = (x∞1)(1∞a), Note that X = for any a ∈ A, x ∈ X. Hence D(X, A) is commutative if and only if X and A are both commutative. From (7), D(X, A) is co-commutative if and only if X and A are both cocommutative. And, from (14) and (11), X is commutative (resp. co-commutative) if and only if A is co-commutative (resp. commutative). Under the condition of Theorem 2.2, A and X are always, respectively, co-commutative and commutative. Hence we get the following: Proposition 2.5 Under the condition of Theorem 2.2, the following are equivalent: (i) D(X, A) is commutative; (ii) A is commutative; (iii) X is co-commutative; (iv) D(X, A) is co-commutative. Corollary 2.6 Assume H is a finite co-commutative perfect weak Hopf algebra with invertible weak antipode. Then the quantum quasi-double D(H) is non-commutative and non-cocommutative if and only if H is non-commutative. For example, for any Clifford monoid S, its quantum quasi-double D(S) is non-commutative and non-co-commutative if and only if S is non-commutative. 3
Almost Weak Antipodes of Quasi-Bicrossed Products
It is known from [1] that for a matched pair (X, A) of Hopf algebras X and A with antipodes SX and SA , respectively, the bicrossed product X A is a Hopf algebra with antipode S given by S(x a) = (x)(a) SA (a ) SX (x ) SA (a ) SX (x ). So, it is natural to ask for X and A satisfying Theorem 2.1, whether X∞A is an almost weak Hopf algebra with an almost weak antipode S given by a similar formula. Unfortunately, in general, it is negative. For example, for a finite Clifford monoid T = {t1 , ..., tn }, the quantum double D(T ) = (kT )op∗ ∞kT of T over a field k, as a quasi-bicrossed product, is a quasi-braided almost −1 −1 bialgebra (see [2]), with t x = x(SkT (t)?t) = x(t−1 ?t) and t x = x(SkT (t)t)t = x(t−1 t)t −1 −1 for t ∈ T and x ∈ (kT )op∗ , where x(SkT (t)?t) means the homomorphism a → x(SkT (t)at) for a ∈ kT ; but, it can be shown that the homomorphism S given by S(x∞a) = (x)(a) SkT (a ) S(kT )op∗ (x )∞SkT (a )S(kT )op∗ (x ) is not an almost weak antipode and then D(T ) = (kT )op∗ ∞ kT is not an almost weak Hopf algebra on S. In fact, for any x ∈ (kT )op∗ and t ∈ T , x (tSkT (t) (S(kT )op∗ (x )x(4) ))∞(tSkT (t) S(kT )op∗ (x )x(5) )t (id ∗ S ∗ id)(x∞t) = (x)
=
(x)
x (tt−1 (S(kT )op∗ (x )x(4) ))∞(tt−1 S(kT )op∗ (x )x(5) )t
Li F.
312
=
x ((S(kT )op∗ (x )x(4) )(tt−1 ?tt−1 ))∞((S(kT )op∗ (x )x(5) )(tt−1 tt−1 ))tt−1 t
(x)
=
x ((S(kT )op∗ (x )x(4) )(tt−1 ?))∞((S(kT )op∗ (x )x(5) )(tt−1 ))t
(x)
(since for a Clifford monoid T, tt−1 is in the centre of T and tt−1 t = t) = x x(4) (tt−1 )S(kT )op∗ (x )x(5) ∞x (tt−1 )x(6) (tt−1 )t = x x (tt−1 )x(4) (tt−1 )S(kT )op∗ (x )x(5) ∞t = x S(kT )op∗ (x (tt−1 )x x(4) (tt−1 ))x(5) ∞t = x S(kT )op∗ (x (tt−1 )x )x(4) ∞t −1 (4) = x x (tt−1 )(x SkT )x ∞t −1 (4) = x (tt−1 )x (x SkT )x ∞t
(since tt−1 in the center of T, x x (tt−1 ) = x (tt−1 )x ) = x (tt−1 )x ∞t. But, for a Clifford monoid T , in general, tt−1 = 1, then x (tt−1 )x = x. So, (id ∗ S ∗ id)(x∞t) = x∞t, i.e. id ∗ S ∗ id = id. Therefore, the homomorphism S given by S(x∞a) = (x)(a) SkT (a )S(kT )op∗ (x )∞SkT (a )S(kT )op∗ (x ) is not an almost weak antipode of D(T ) = (kT )op∗ ∞kT . However, we have found some cases of matched pairs (X, A) of weak Hopf algebras X and A with antipodes SX and SA , respectively, whose bicrossed products X∞A are weak Hopf algebras with antipodes S given by S(x∞a) = (x)(a) SA (a ) SX (x )∞SA (a ) SX (x ) or a substituted formula. See the following: Proposition 3.1 Suppose (X, A) is a quasi-matched pair of bialgebras where X is a left Hopf algebra with left antipode SX and A is a weak Hopf algebra with weak antipode SA . Then the quasi-bicrossed product X∞A is an almost weak Hopf algebra with an almost weak antipode S given by S(x∞a) = (x)(a) SA (a ) SX (x )∞SA (a ) SX (x ) for any x ∈ X, a ∈ A. Proof We have known that X∞A is an almost bialgebra. So, it is enough to show that S is an almost weak antipode. For x ∈ X and a ∈ A, (id ∗ S ∗ id)(x∞a) = (x ((a SA (a(4) )) SX (x ))∞(a SA (a )) SX (x ))(x(4) ∞a(5) ) (x)(a)
=
x ((a (SA (a(6) )) SX (x(4) ))((a SA (a(5) ) SX (x )) x(5) )
(x)(a)
∞(a SA (a(4) )) (SX (x )x(6) )a(7) ) = x (a SA (a(4) ) SX (x )x(4) )∞(a SA (a ) SX (x )x(5) )a(5) (x)(a)
=
x (a SA (a(4) ) ε(x ))∞(a SA (a ) SX (x )x(4) )a(5)
(x)(a)
=
(x)(a)
x ε(a SA (a(4) ))∞(a SA (a ) SX (x )ε(x )x(4) )a(5)
On Quasi-Bicrossed Products of Weak Hopf Algebras
=
313
x ∞ε(x )a SA (a )a = id(x∞a).
(x)(a)
So, id ∗ S ∗ id = id. Similarly, it can be shown that S ∗ id ∗ S = S. Hence S is an almost weak antipode. Similarly, we have: Proposition 3.2 Suppose (X, A) is a quasi-matched pair of bialgebras where X is a weak Hopf algebra with weak antipode SX and A is a right Hopf algebra with right antipode SA . Then the quasi-bicrossed product X∞A is an almost weak Hopf algebra with an almost weak antipode S = T ∗ id ∗ T , where T satisfies T (x∞a) = (x)(a) SA (a ) SX (x )∞SA (a ) SX (x ) for any x ∈ X, a ∈ A. Firstly, we give a relation on T . For x ∈ X and a ∈ A, (id ∗ T ∗ id)(x∞a) = x (a SA (a(4) ) SX (x )x(4) )∞(a SA (a ) SX (x )x(5) )a(5)
Proof
(x)(a)
=
x (a SA (a ) SX (x )x(4) )∞(ε(a ) SX (x )x(5) )a(4)
(x)(a)
=
x (a SA (a ) SX (x )x(4) )∞ε(a )ε(SX (x ))ε(x(5) )a(4)
(x)(a)
=
x (ε(a ) SX (x )x )∞a
(x)(a)
=
x (ε(a ) SX (x )x )∞a
(x)(a)
= id(x∞a). So, id ∗ T ∗ id = id. Then, it is easy to see that S ∗ id ∗ S = S and id ∗ S ∗ id = id. Hence S is an almost weak antipode of X∞A. Note that in Proposition 3.2, in general, T ∗ id ∗ T = T . Hence T itself is not an almost weak antipode. It is easy to see that a sufficient condition for S ∗ id ∗ S = S is a x ∞a x = a x ∞a x , for any x ∈ X and a ∈ A. As is well known (see [4]), for a k-algebra L and a k-bialgebra K, if L is a left K-module algebra with action , then we can define the smash product (also called the semi-direct product in [4]) L#K of L and K with its multiplication satisfying (x#a)(y#b) = (a) x(a y)#a b for x, y ∈ L and a, b ∈ K and its identity 1#1. In this case, L#K is an algebra. Moreover, if L is a left K-module-bialgebra and K is co-commutative, then L#K becomes a bialgebra with ∆(x#a) = (a)(x) (x #a ) ⊗ (x #a ) and ε(x#a) = ε(x)ε(a). It is interesting to note that in the definition of a bicrossed product, supposing X is a left A-module-bialgebra, A is co-commutative and a x = aε(x) for a ∈ A, x ∈ X (A is a right X-module-co-algebra obviously), then X∞A is, in fact, the smash product X#A of X and A and x∞a = x#a for x ∈ X, a ∈ A. Therefore, for a co-commutative bialgebra K and a left K-module-bialgebra, the smash product L#K is their special bicrossed product. About (almost) weak antipodes of L#K, we have the following: Theorem 3.3
Suppose that L is a weak Hopf algebra with weak antipode SL , K is a co-
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commutative Hopf algebra with antipode SK and L is a left K-module-bialgebra with the action . Then L#K is a weak Hopf algebra with weak antipode S satisfying S(x#a) = (a) (SK (a ) SL (x))#SK (a ) for x ∈ L, a ∈ K. In this situation, L#K is co-commutative iff L is cocommutative. Proof Under these conditions, we have known that L#K is a bialgebra. So, we need to prove only that S is a weak antipode. For x ∈ L, a ∈ K, we have (S ∗ id ∗ S)(x#a) = S(x #a )(x #a )S(x #a ) =
(a)(x)
(SK (a ) SL (x )#SK (a ))(x #a )((SK (a(5) ) SL (x ))#SK (a(4) ))
(a)(x)
=
((SK (a ) SL (x ))(SK (a ) x )#SK (a )a(4) )((SK (a(6) ) SL (x ))#SK (a(5) ))
(x)(a)
=
((SK (a ) SL (x )x )#SK (a )a )((SK (a(5) ) SL (x ))#SK (a(4) ))
(x)(a)
=
(SK (a ) SL (x )x )((SK (a )a ) (SK (a(5) ) SL (x )))#(SK (a )a ) SK (a(4) )
(x)(a)
=
(SK (a ) SL (x )x )(SK (a )a(4) )(SK (a(7) ) SL (x ))#SK (a )a(5) SK (a(6) )
(x)(a)
=
(SK (a ) SL (x )x )(SK (a ) SL (x ))#SK (a )
(x)(a)
=
(SK (a ) SL (x )x SL (x ))#SK (a )
(x)(a)
=
(SK (a ) SL (x))#SK (a ) = S(x#a); (a)
(id ∗ S ∗ id)(x#a) = =
(x #a )S(x #a )(x #a )
(x)(a)
(x #a )((SK (a ) SL (x ))#SK a )(x #a(4) )
(x)(a)
=
(x (a (SK (a(4) ) SL (x )))#a SK (a ))(x #a(5) )
(x)(a)
=
(x (a SK (a(4) ) SL (x )))((a SK (a )) x )#(a SK (a )) a(5)
(x)(a)
=
(x (a SK (a(6) ) SL (x )))(a SK (a(5) ) x )#(a SK (a(4) ))a(7)
(x)(a)
=
(x ε(a ) S(L) (x ))(ε(a ) x )#a = x#a.
(x)(a)
Hence S ∗ id ∗ S = S and id ∗ S ∗ id = id. Then SK is a weak antipode. From the definition of the co-multiplication ∆, it is easy to see that L#K is co-commutative iff L is co-commutative. Note that this result is a generalization of the result in Section 2.4 in [4]. In the proof, we only need SK as a right antipode. However, a co-commutative right Hopf algebra is always a
On Quasi-Bicrossed Products of Weak Hopf Algebras
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Hopf algebra. So, in Theorem 3.3, we have to claim K to be a Hopf algebra. As mentioned above, the smash product L#K is a special one of the (quasi-)bicrossed products. Hence this theorem, Propositions 3.1 and 3.2 give some partial positive answers for the problem on almost weak Hopf algebra structures of quasi-bicrossed products in certain special cases. The following seems be a light on our subsequent research: Example 3.4 For a regular monoid S and a group G, let ρ be a monoid morphism from G to End0 (S) (the set of all endomorphisms f of S satisfying f (1) = 1). Define an action of G on S with x a = ρ(x)a, for any x ∈ G, a ∈ S. Then kS becomes a left kG-module-bialgebra. By Theorem 3.3, kS#kG is a co-commutative weak Hopf algebra. Set S ×ρ G = {a#x : a ∈ S, x ∈ G}. It can be proved that S ×ρ G is a regular monoid under the multiplication of kS#kG. One calls S ×ρ G a smash product (or, semi-direct product) of a regular monoid S and a group G (see [7]). It is easy to see that L#K = k(S ×ρ G) and the monoid of all group-like elements G(L#K) = S ×ρ G. 4
Some Properties on Quantum Quasi-Doubles
As seen above, a quasi-bicrossed product D(X, A) in Theorem 2.2 is just a quantum quasidouble in the case of X or A being finite dimensional. We will discuss it further. Firstly, we relate D(H) and D(H opcop∗ )op by an isomorphism. Define a quasi-R-morphism f from a quasi-braided almost bialgebra (H, R) to another (H , R ) to be an almost bialgebra map f : H −→ H satisfying R = (f ⊗ f )(R).
Theorem 4.1 Suppose that H is a finite-dimensional co-commutative perfect weak Hopf algebra with invertible weak antipode T . For a basis {e1 , ..., en } of H and the dual basis {e∗1 , ..., e∗n } of ¯ defined by ϕ(p ⊗ h) = h ⊗ p for p ∈ H ∗ and H ∗ , the map ϕ : (D(H), R) −→ (D(H opcop∗ )op , R), n ¯ = n (ei ∞ε)⊗(1∞e∗ ). h ∈ H, is a quasi-R-isomorphism with R = i=1 (ε∞ei )⊗(e∗i ∞1), R i i=1 Proof
As co-algebras, we have
D(H opcop∗ )op = D(H opcop∗ ) = (H opcop∗ )∗cop ∞H opcop∗ = H op ∞H ∗opcop = H∞H ∗cop and D(H) = H ∗cop ∞H. Then it is trivial that ϕ is a co-algebra morphism. Now, note that H opcop∗=H ∗opcop and (H opcop∗ )∗cop=H op are sub-algebras of D(H opcop∗ )∗op and the weak antipodes of H ∗opcop and H op are T ∗−1 and T −1 (see [2]). Denote by •op the multiplication of D(H opcop∗ )op . Then by Lemma 2.4, for p, q ∈ H ∗cop , h, e ∈ H, we have −1 ϕ((p∞h)(q∞e)) = ϕ q , h q , T (h )pq ∞h e (q)(h)
=
e•op h ∞h , q T −1 (h ), q q •op p
(q)(h)
=
e•op h ∞h , q h , T ∗−1 (q )q •op p
(q)(h)
= (e∞q)(h∞p) = (h∞p)•op (e∞q) = ϕ(p∞h)•op ϕ(q∞e). Therefore ϕ is an algebra morphism. Clearly, ϕ is bijective. Hence it is an almost bialgebra isomorphism.
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¯ Then R ¯ is a quasi-R-matrix of D(H opcop∗ )op And, (ϕ ⊗ ϕ)R = i (ei ∞ε) ⊗ (1∞e∗i ) = R. since ϕ is an isomorphism. It means that ϕ is a quasi-R-morphism from (D(H), R) to (D(H opcop∗ )op , R). The result above is a generalization of the correspondent theorem on a quantum double in [8]. Now, we discuss the representation-theoretic interpretation of D(H). Definition 4.2 (See [1]) For a bialgebra H over k, a crossed H-bimodule V is a vector space together with linear maps µV : H ⊗ V −→ V and ∆V : V −→ V ⊗ H such that (i) The maps µV and ∆V turn V into a left H-module and a right H-comodule, respectively ; (ii) (a)(β) a βV ⊗ a βH = (a)(a β) (a β)V ⊗ (a β)H a for all a ∈ H and β ∈ V ; where we set µV (a ⊗ β) = aβ and ∆V (β) = (β) βV ⊗ βH . Theorem 4.3 Suppose H is a finite-dimensional co-commutative perfect weak Hopf algebra with invertible weak antipode T . Then : (i)
Any left D(H)-module has a natural structure as a crossed H-bimodule;
(ii)
For a crossed H-bimodule V , if T −1 (a )a βH ⊗ a βV = βH ⊗ aβV (a)(β)
(18)
(β)
for all a ∈ H and β ∈ V , then V has a natural structure as a left D(H)-module. ∼ H ∗cop ∞1 and H ∼ Proof (i) Let V be a left D(H)-module. Since H ∗cop = = ε∞H are ∗ subalgebras of D(H), V is a left H-module and also a left H -module with aβ = (1∞a)β and xβ = (x∞1)β for a ∈ H, x ∈ H ∗ , β ∈ V . Then, (ax)(β) = a(xβ). But, from Lemma 2.4, we get ax = (1∞a)(x∞1) = x, T −1 (a )?a ∞a = (x, T −1 (a )?a ∞1)(1∞a ). So, a(xβ) =
(a)(x) x, T
(a)(x) −1
(a)(x)
(a )?a (a β).
One must show that V can be endowed with a crossed H-bimodule structure. For µV , we define µV (a ⊗ β) = aβ. How do we define ∆V in Definition 4.2? Given a basis {e1 , . . . , en } of H and the dual basis {e∗1 , . . . , e∗n } of H ∗ , note that x = n ∗ ∗ ∗ i=1 x(ei )ei and a = i=1 ei (a)ei , for x ∈ H , a ∈ H. Define ∆V : V −→ V ⊗ H satisfying ∆V (β) = i e∗i ⊗ ei , for any β ∈ V . Consider the duality ∆∗V of ∆V , we have that for any α ∈ V ∗ , β ∈ V , x ∈ H ∗ , n n ∆∗V (α ⊗ x), β = α ⊗ x, ∆V (β) = α, e∗i βx, ei = α, x, ei e∗i β = α, xβ; n
i=1
i=1 ∗
in particular, for x = ε (the identity of H ) and any β ∈ V , ∆∗V (α ⊗ ε), β = α, β. Then, ∆∗V (α ⊗ ε) = α. It follows that, for any y ∈ H ∗ , ∆∗V (∆∗V (α ⊗ x) ⊗ y), β = ∆∗V (α ⊗ x), yβ = α, x(yβ) = ∆∗V (α ⊗ xy), β; then ∆∗V (∆∗V (α ⊗ x) ⊗ y) = ∆∗V (α ⊗ xy). Hence V ∗ is a right H ∗ -module under the action ∆∗V . Therefore, dually, V becomes a right H-co-module under the co-action ∆V .
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For a ∈ H, β ∈ V , x ∈ H ∗cop , we have n a βV ⊗ a βH = a (e∗i β)x(a ei ) (id ⊗ x) (a) i=1
(a)(β)
=
n
a (e∗i β)x (ei )x (a ) =
(a)(x) i=1
=
(a)(x)
(4)
x (a
)x (T
−1
(a )?a )(a β) =
(a)(x)
=
=
a (x β))x (a )
x(?a T
−1
(4)
(a )a )(a
β) =
(a)
x(?a )(a β) =
(a)
(a)
n
n
x (a )x (ei )e∗i (a β) =
(a)(x) i=1
= (id ⊗ x)
n
e∗i (a β) ⊗ ei a
It follows that (a)(β) a βV ⊗ a βH = V is a crossed H-bimodule.
x (a )x (a β)
(a)(x)
x(ei a )e∗i (a β)
(a) i=1
(a) i=1
x(a(4) T −1 (a )?a )(a β)
= (id ⊗ x)
(a β)V ⊗ (a β)H a .
(a β)(a)
(a β)(a) (a
β)V ⊗ (a β)H a . Hence by Definition 4.2,
(ii) Note that V is a crossed H-bimodule about µV and ∆V . Then, V is a left H-module about µV and a right H-co-module about ∆V . Write µV (a ⊗ β) = aβ, for a ∈ H, β ∈ V . For x ∈ H ∗ , β ∈ V , let xβ = (β) x, βH βV , where ∆V (β) = (β) βV ⊗ βH . Since ∆V is a right co-action, it is easy to show that (xy)β = x(yβ) for y ∈ H ∗ . Then it follows that V is a left H ∗cop -module. Set (xa)β = x(aβ), for x ∈ H ∗cop , a ∈ H, β ∈ V . Then, a(xβ) = x, βH aβV = x, T −1 (a )a βH a βV (β)
=
(a)(β)
x, T −1 (a )(a β)H a (a β)V
(a)(a β)
=
x , T −1 (a )x , (a β)H x , a (a β)V
(a)(x)(a β)
=
x , T −1 (a )x , a x (a β) =
(a)(x)
x, T −1 (a )?a (a β) (a)
−1 −1 = x, T (a )?a a β = x, T (a )?a ∞a β = (ax)β. (a)
(a)
It follows that V becomes a left D(H)-module since H and H ∗cop are subalgebras of D(H) and the multiplication of D(H) is determined by the interaction of H and H ∗cop . At the end, let us look at the regularity of the quasi-R-matrices of quantum quasi-doubles. It is well known that the R-matrix of a quantum double of a finite-dimensional Hopf algebra (in particular, a finite group) is invertible. However, generally, for a finite-dimensional cocommutative perfect weak Hopf algebra (in particular, a finite Clifford monoid), the quasi-Rmatrix of its quantum quasi-double is not invertible. In [2], we show only that for the quantum quasi-double of a finite Clifford monoid, the quasi-R-matrix is partially invertible. In [6], we
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proved that the quasi-R-matrix R of every quasi-braided almost weak Hopf algebra is regular ¯ such that RRR ¯ = R, RR ¯ R ¯ = R. ¯ Hence, if a quantum in von Neumann’s sense, i.e. there is R quasi-double has a weak antipode so as to become a quasi-braided almost weak Hopf algebra, then its quasi-R-matrix must be regular. But, we have known in Section 4, in general, one cannot find an almost weak antipode in the natural way for a quantum quasi-double. In this situation, we have to consider the regularity of the quasi-R-matrix of a quantum quasi-double without any aid of its possible weak antipodes as follows: Theorem 4.4 For a finite-dimensional co-commutative perfect weak Hopf algebra H with n inverse weak antipode T , the quasi-R-matrix R = i=1 (ε∞ei ) ⊗ (e∗i ∞1) of its quantum quasi¯ = n (ε∞ei )⊗(e∗ T ∞1), double D(H) is a regular element in D(H)⊗D(H) with its inverse R i i=1 where {e1 , ..., en } is a basis of H and {e∗1 , ..., e∗n } is the dual basis in H∗ . For any ξ = b ⊗ u ⊗ c ⊗ v ∈ H ⊗ H ∗ ⊗ H ⊗ H ∗ , we have ¯ ξ = ε(b)u(ei ej el )(e∗i (e∗j T )e∗l )(c)v(1) RRR,
Proof
i,j,l
= ε(b)v(1)
n n n u ei e∗i (c ) ej e∗j (T (c )) el e∗l (c ) i=1
(c)
j=1
l=1
n = ε(b)v(1)u c T (c )c = ε(b)v(1)u(c) = ε(b)v(1) u(ei )e∗i (c) = R, ξ; (c)
i=1
¯ = R. Similarly, RR ¯ R, ¯ ξ = R, ¯ ξ; then RR ¯ R ¯ = R. ¯ then RRR R R For a left D(H)-module V , define CV,V satisfying CV,V (v ⊗ w) = τ (R(v ⊗ w)) for v, w ∈ V ; R then CV,V is a solution of the classical Yang–Baxter equation, where τ is the flip map defined as τ (v1 ⊗ v2 ) = v2 ⊗ v1 . From Theorem 4.4, it is easy to prove that Corollary 4.5 For a finite-dimensional co-commutative perfect weak Hopf algebra H with R inverse weak antipode T , let V be a left D(H)-module. Then CV,V is regular in the endomor¯ ¯ R R ¯ ⊗ w)) for v, phism monoid of V ⊗ V , with its inverse CV,V satisfying CV,V (v ⊗ w) = τ (R(v w ∈V. Acknowledgements This paper was prepared at the University of Tasmania with the support of an Australia Research Council Grant, thanks to Prof. Peter Trotter’s help the author very much appreciates the referee’s kind advice. References [1] Kassel, C.: Quantum Groups, Springer-Verlag, New York, 1995 [2] Li, F.: Weak Hopf algebras and some new solutions of quantum Yang–Baxter equation. J. Algebra, 208, 72–100 (1998) [3] Takeuchi, M.: Some topics on GLq(n). J. Algebra, 147, 379–410 (1992) [4] Abe, E.: Hopf Algebras, Cambridge University Press, Cambridge, (1980) [5] Green, J. A., Nicholes, W. D., Taft, E. J.: Left Hopf algebras. J. Algebra, 65, 399–411 (1980) [6] Li, F.: Solutions of Yang–baxter equation in endomorphism semigroups and quasi-(co)braided almost bialgebras. Comm. Algebra, 28(5), 2253–2270 (2000) [7] Nico, W. R.: On the regularity of semidirect products. J. Algebra, 80, 29–36 (1983) [8] Radford, D. E.: Minimal quasi-triangular Hopf algebras. J Algebra, 157, 285–315 (1993)
