Self-duality and generalized Bicrossproducts Hopf algebras

Group factorizations are very common in mathematics. Among their uses is the bicrossproduct construction which is one of the primary sources of non-commutative and noncocommutative Hopf algebras. These bicrossproduct Hopf algebras have been introduced by Majid10 and Takeuchi16. Since then, bicrossproduct Hopf algebras have been extensively studied2-4,6,9. These algebras have many applications, for example Majid in10 showed that they can be considered as a systems combine quantum mechanics with geometry11.


INTRODUCTION
Group factorizations are very common in mathematics. Among their uses is the bicrossproduct construction which is one of the primary sources of non-commutative and noncocommutative Hopf algebras. These bicrossproduct Hopf algebras have been introduced by Majid 10 and Takeuchi 16 . Since then, bicrossproduct Hopf algebras have been extensively studied 2-4,6,9 . These algebras have many applications, for example Majid in 10 showed that they can be considered as a systems combine quantum mechanics with geometry¹¹.
In 1996, Beggs et al., 5 have computed the quantum double construction of Drinfeld 7 for the bicrossproduct Hopf algebra associated to the factorization X = GM, where G and M are subgroups of the group X, which led to an interesting generalization of crossed modules to bicrossed bimodules. In addition, they showed that basispreserving selfduality structures for the bicrossproduct Hopf algebras are in one-to-one correspondence with factor-reversing group isomorphisms.
kG becomes a ring. The map kG : k kG given by kG(a) = a1 where 1 is the identity element of G makes kG a k-algebra. The k-algebra kG is said to be the semigroup k-algebra of G 1 ... (1) Associating to this factorization, we can define the bicrossproduct Hopf algebra H =kM  k(G) with basis s u where s  M and u  G. The product, unit, coproduct, counit and antipode are defined as follows 5 : Also, we can define the dual of H which is again a bicrossproduct Hopf algebra H *=k(M)  kG with basis  s u where s  M and u  G. The product, unit, coproduct, counit and antipode are defined as follows 5

Self-duality of bicrossproducts
Here we study the bicrossproduct Hopf algebras associated to a factorization of a group into a subgroup and a semisubgroup with identity and a left inverse property. This may have some relevance to the work of Green, Nichols and Taft 8 concerning one sided Hopf algebras structures. If it exists, the left inverse for an element a  H will be denoted by a L . Let X = GH be a group which factorizes into a subgroups G and a semisubgroupwith identity H. Then H acts on G through the right action It can be seen that we can associate to this factorization a bicrossproduct bialgebra H = kH  k(G) with basis au where a  H and u  G. The product, unit, coproduct and counit are defined as follows: If H posses a left inverse a L for each a  H, then H becomes a Hopf algebra and the antipode will be given by: Due to these formulas, it can be noted that H = kH  k(G) has the smash product algebra structure by the induced action of H and the smash coproduct coalgebra structure by the induced coaction of G.
In the symbol H = kH  k(G), kH is the semigroup Hopf algebra of the semigroup H with identity and left inverse property. A basis of kH is given by the elements of H, with multiplication given by the semigroup product in H , and comultiplication given by a = aa for a  H. Also, k(G) is the Hopf algebra of functions on G with basis given by  u for u  G . The product is just multiplication of functions, and the coproduct is Moreover, the  par t means that kH acts on k(G), and the  part means that k(G) coacts on kH.
In addition, a dual bicrossproduct bialgebra H * = k(H)  kG can be defined with basis  a u where aH and uG. The product, unit, coproduct and counit are defined as follows: If H posses the left inverse property for each aH, then H * becomes a Hopf algebra and the antipode will be given by: It can be noted that what has been said about H, can be dually said about H *.

Definition 3.1
Let X be a finite group and X = GH be factorization of X into two subsemigroups G and H with identities. A semigroup isomorphism f : XX is defined to be factor-reversing if (g)H for all g G and (a)G for all a H. We need the following lemmas:

Lemma 3.2
Let X = GH be factorization of a group X into a subgroup G and a subsemigroup H with identity. Then for the algebra H = kH  k(G) ,where k(G) is the algebra of function on G and kH is the semigroup algebra of H , an algebra homomorphism: H H * which sends basis elements to basis elements can be constructed from afactor-reversing isomorphism of X = GH.

Proof
We Suppose that  is a semigroup isomorphism and we define a linear map  : We want to prove that f is an algebra By the uniqueness of factorization, we have ... (5) Now to prove that f is an algebra for a u b u H , a, b H and u   G. We start with the left hand side as follows: , , Now, the question arises "does the same result hold for the coalgebra". The answer is in negative as the counit property is not applicable unless we assume that our semigroup H posses, at least, the left inverse property as we see in the following lemma.

Lemma 3.3
Let X = GH be factorization of a group X into a subgroup G and a subsemigroup with identity and left inverse property H. Then for the coalgebra H = kH  k(G) ,where k(G) is the algebra of function on G and kH is the semigroup algebra of H, there is a coalgebra homomorphism: H H * which sends basis elements to basis elements can be constructed from a factor-reversing isomorphism of X = GH.

Proof
We suppose that We have used the assumption that f is a semigroup homomorphism. Also, we get

Theorem 3.4
Let X = GH be factorization of a group X into a subgroup G and a subsemigroup with identity and left inverse property H. Then for the Hopf algebra H = kH  k(G), where k(G) is the algebra of function on G and kH is the semigroup algebra of H, there is a Hopf algebra isomorphism: HH * which sends basis elements to basis elements can be constructed from a factor-reversing isomorphism of X = GH.

Proof
We Suppose that where aH and uG. The conditions for  to be an algebra and a coalgebra isomorphism follow from lemmas 3.2 and 3.3. To prove that ef is a Hopf algebra isomorphism, we need to check the antipode property and the inevitability of f . First, we need the following calculations: By the uniqueness of factorization, we get ... (8) Due to the fact that f is a semigroup isomorphism, we get  (11) and show that : where id is the identity map, as follows The third equality is due to the identities with the fact that f is an isomorphism. Also we have as required. Therefore,  is a Hopf algebra isomorphism.
Following theorem reveals that the converse of Theorem 3.4 is also true.

Theorem 3.5
Let X = GH be factorization of a group X into a subgroup G and a subsemigroup with identity and left inverse property H. Then the factor-reversing isomorphisms of X = GH give rise to Hopf algebra self-duality pairings <,> : H Hk on the Hopf algebra H = kH elements of our two Hopf algebras, and we want to prove that we can induce a group isomorphism  -1 from  -1 . We start with functions h : H × GH and g : H × GG given by ... (12) As  is an algebra isomorphism, it preserves the unit and the product. Starting with the unit, we get ... (13) but, since f is an algebra isomorphism we have ... (14) for some sH. Comparing equations (13) and (14)  Also, from the coproduct formula, we get g(a  x, y) = g(a, u) with u = xy, i.e., g(a  x,y) = g(a, xy). Putting y = e gives combining equations (32)  is a group homomorphism. It can be noted that our Hopf algebra map  -1 is certainly that one obtained by f -1 , which is well defied due to G H = {e}. Since  -1 is a Hopf algebra isomorphism, it is invertible.
So if we put it can be easily shown that  is obtained by the group isomorphism by using a similar technique.