A Generalization for -Cocycles
We will give generalized definitions called type II -cocycles and weak quasi-bialgebra and also show properties of type II -cocycles and some results about weak quasi-bialgebras, for instance, construct a new structure of tensor product algebra over a module algebra on weak quasi-bialgebras.
We will introduce a new generalized definition called type II -cocycle; the namely, relax the invertible condition of associator in -cocycle up to adding an associator satisfies all forms in definition of -cocycle together with the original associator, and both need not to be invertible for each other; then we give examples to illustrate it clearly. Majid have shown many results about -cocycle in , and we obtain some results including cohomologous concept through this new definition, main properties of type II -cocycles, and its simple classification.
It is well known that quasi-bialgebras and quasi-Hopf algebras play important roles in quantum group theory, and these concepts were introduced by Drinfel’d in  who relaxed the coassociative law of up to conjugation. In this paper, we will show a new definition called weak quasi-bialgebra, a generalization of quasi-bialgebras, and there are simple examples to illustrate. Authors show results for weak quasi-bialgebras in place of quasi-bialgebras (cf. [1, 3]), including that there exists an algebra structure on , a generalization of the algebra product in , where is a weak quasi-bialgebra and is a -module algebra.
We follow all the notation and conventions in , throughout the paper. In the following, will be a fixed field throughout, and all algebras, coalgebras, vector spaces, and so forth are over automatically unless specified. We recall definitions as follows.
Definition 1.1. For any bialgebra, if there is an invertible element such that where integers and are max even number and max odd number, respectively, in , we call an -cocycle. If (), then the cocycle is counital. We define , ,, and .
Definition 1.2. Let be a algebra with unit and homomorphisms , . If there exists a counital 3-cocycle rendering that, for all ,
Then is called a quasi-bialgebra together with coproduct and counit , and call an associator.
We will denote the tensor components of with big and small letters, respectively, for instance, where is the ith factor.
Definition 1.3. Let be a quasi-bialgebra and a vector space. If has a multiplication and the unit obeying that
for any and , where is the module structure map of , then say is a left module algebra.
In bialgebras, the composition of any two algebra homomorphisms satisfies the equality for all . We will use this equation frequently.
2. Type II Cocycles and Weak Quasi-Bialgebras
Definition 2.1. An associative algebra with unit over a commutative ring called a fake bialgebra, if there are two algebra homomorphisms and .
Definition 2.2. Let be a fake bialgebra, and elements . Denote
where integers and in are max even number and max odd number, respectively. If there exists an element with , which makes and , satisfy all equalities that one side of equalities is not a single item at least, similar to (1.1). Then call the pair a type II -cocycle for and denote it by . The cocycle is called counital, if both and are counital.
There is a nature way to define type I cocycle similarly. If we require the type II -cocycle for to satisfy all transformations of (1.1), but each side of formulas must have one item at least, then the type II -cocycle is called type I -cocycle and denoted by .
We write , , and briefly for , , and without confusions, respectively. To clarify a new definition above, we give simple examples on a fake bialgebra . In the following, we discuss type II -cocycles only.
Example 2.3. Cocycle means both and are in , obeying that And there is such that where .
Example 2.4. Cocycle and , , where , satisfy that
Observing examples, we can see that and are replaced by and , respectively, after moving to a corresponding place in the other side of equations, and vice versa. Obviously, -cocycle must be a type II 2-cocycle and .
Example 2.5. Let be an associative algebra with an idempotent over a field . Define by and by , for all . It is clear that is a fake bialgebra. We set and , then . It is easy to check that is a type II 1-cocycle.
Proposition 2.6. Let be a cocycle for a fake bialgebra , and denote where integers and in are max even number and max odd number, respectively. Then one has the following.(1), and . (2)If is commutative with , then . Especially, if either or is zero, then the other one is zero too. On the other hand, if either of elements and is not zero, then the rest elements in set are not zero. (3)If is a left unit and is not a right zero divisor, then . (4)If has a right inverse , so do , , and . Similarly, if has a left inverse , so do , , and .
Proof. We obtain that by
since is a homomorphism. And
Analogously, we have that
Then, we get easily that and
Finally, there is the equality that
We, last, compute that
Therefore and are idempotent.
Obviously, we get this by statement . Let and assume , from the previous part, that yields to contradicting . Therefore must be zero, and for the same reason.
The equality suggests that , and then . It is clear that because . In addition, implies that As a result, we have if is not a right zero divisor.
It is easy to obtain that and as and , respectively. But then and has a right inverse. Likewise, we can prove the rest part.
Furthermore, if or has a one-side inverse, it makes sense that since both and are idempotent. We also have that which indicates is a left unit and a right unit, by if . Hence and cannot be anything but the identity element if one of them is a one-side unit.
Corollary 2.7. The following statements are equivalent.(1)Φ has a right inverse.(2).(3)ϕ has a left inverse. (4)∂ is a one-side unit.(5)χ is a one-side unit.(6)ψ has a left inverse.
Proof. (Sketch of Proof).
Check by .
Equality suggests that classification of is divided into three types. The first type , and the second if , that is, . The last one is that is a right zero divisor.
Example 2.8. In algebra over the integer ring , we define given by and for any , and by for all such that is a fake bialgebra. Set such that . The product of any two elements in the set equals , obviously. We also set ; then, it is easy to prove that is a cocycle and a right zero-divisor .
Proposition 2.9. Let be a fake bialgebra with counital law of . If and (, resp.), then (, resp.) is counital if and only if (, resp.).
Proof.. Since that , we have rendering that if and only if , where .
Proposition 2.10. Let be a fake bialgebra with coassociative law of and there are elements in . If obeys , then . Especially, is a 2-cocycle if is invertible if is a bialgebra.
Proof.. To obtain the result, we observe that
We have known that is cohomologous to for a bialgebra if is a counital 2-cocycle, which was mentioned by Majid in . Let be a bialgebra, and cocycle for . Denote that and . Then we have the following.
Proposition 2.11. If equality holds and is a commutative element in set , and commutes with any element in set as well, then
Proof. A long equality showed that
There exists a similar version for , namely, the following preposition.
Proposition 2.12. If there is the equation and commutes with any element in set , and is a commutative element in set as well, then
Proposition 2.13. Let be a bialgebra and a counital cocycle for , and define for all , then the algebra with original and consists a new coalgebra if . Moreover, is an algebra map if , then algebra is a bialgebra with comultiplication .
Proof. It is clear that . So we only need to show the coassociative law of . For all , we obtain Finally, for any ,
Definition 2.14. Let be a fake bialgebra. If there exists a cocycle for obeying that for all , then is called a weak quasi-bialgebra.
Example 2.15. Let be an associate algebra over field , where the characteristic of is not 2. And is a dimensional vector space with basis obeying that , and . We define homomorphisms given by and given by . Obviously, is a fake bialgebra. Set and , then is a cocycle with holding . It is routine to check is a weak quasi-bialgebra.
We relax Definition 1.3 by setting that is a weak quasi-bialgebra so that we can define an algebra structure on , if is a left module algebra and a weak quasi-bialgebra, given by for all , while is equal to here.
Theorem 2.16. Let be a weak quasi-bialgebra and a left module algebra. Then is an associative algebra under the multiplication mentioned above and is the unit.
If is an inverse of , then the multiplication becomes that which is as exact as the one in .
The authors were supported by Guangxi Science Foundation (2011GXNSFA01844), the Scientific Research Foundation of Guangxi Educational Committee (200911 MS145), and 2012 Guangxi New Century Higher Education Teaching Reform Project (2012JGA162). The authors would like to thank the referee for precious suggestions.
V. G. Drinfel'd, “Quasi-Hopf algebras,” Leningrad Mathematical Journal, vol. 1, no. 6, pp. 1419–1457, 1989.View at: Google Scholar