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Semientwining Structures and Their Applications
Semientwining structures are proposed as concepts simpler than entwining structures, yet they are shown to have interesting applications in constructing intertwining operators and braided algebras, lifting functors, finding solutions for Yang-Baxter systems, and so forth. While for entwining structures one can associate corings, for semientwining structures one can associate comodule algebra structures where the algebra involved is a bialgebra satisfying certain properties.
1. Introduction and Preliminaries
Quantum groups appeared as symmetries of integrable systems in quantum and statistical mechanics in the works of Drinfeld and Jimbo. They led to intensive studies of Hopf algebras from a purely algebraic point of view and to the development of more general categories of Hopf-type modules (see  for a recent review). These serve as representations of Hopf algebras and related structures, such as those described by the solutions to the Yang-Baxter equations.
Entwining structures were introduced in  as generalized symmetries of noncommutative principal bundles and provide a unifying framework for various Hopf-type modules. They are related to the so-called mixed distributive laws introduced in .
The Yang-Baxter systems emerged as spectral-parameter independent generalization of the quantum Yang-Baxter equation related to nonultra-local integrable systems [4, 5]. Interesting links between the entwining structures and Yang-Baxter systems have been established in [6, 7]. Both topics have been a focus of recent research (see, e.g., [8–13]).
In this paper, we propose the concepts of semientwining structures and cosemientwining structures within a generic framework incorporating results of other authors alongside ours. The semientwining structures are some kind of entwining structures between an algebra and a module which obey only one-half of their axioms, while cosemientwining structures are kind of entwining structures between a coalgebra and a module obeying the other half of their axioms. The main motivations for this terminology are the new constructions which require only the axioms selected by us (constructions of intertwining operators and Yang-Baxter systems of type II or liftings of functors), our new examples of semientwining structures, simplification of the work with certain structures (Tambara bialgebras, lifting of functors, braided algebras, and Yang-Baxter systems of type I), the connections of the category of semientwining structures with other categories, and so forth. Let us observe that while for entwining structures one can associate corings, for semientwining structures one can associate comodule algebra structures provided the algebra involved is a bialgebra with certain properties (see Theorem 9).
The current paper is organised as follows. Section 2 contains the newly introduced terminology with examples, new results, and comments. Section 3 is about some of the applications of these concepts, namely, new constructions of intertwining operators and braided algebras, lifting functors, and the presentations of Tambara bialgebras and of (new families of) Yang-Baxter systems (of types I and II).
The main results of our paper are Theorems 19, 22, 24, 40, and 41. Theorems 29 and 31 are mentioned in the context of stating some of our results. Theorem 34 is used to prove Theorem 40, while Theorem 37 is related to Theorem 38.
Unless otherwise stated, we work over a commutative ring . Unadorned tensor products mean tensor products over .
For any -module , denotes tensor algebra of . In section 3.5, we work over a field . For an -module, we denote by the identity map. For any -modules and we denote by the twist map, defined by . Let be an -linear map. We use the following notations: , , .
Definition 1. An invertible -linear map is called a Yang-Baxter operator if it satisfies
Remark 2. Equation (1) is usually called the braid equation. It is a well-known fact that the operator satisfies (1) if and only if satisfies the quantum Yang-Baxter equation (if and only if satisfies the quantum Yang-Baxter equation):
2. Semientwining Structures and Related Structures
Definition 3 (Semientwining Structures). Let be an -algebra, and let be an -module, then the -linear map is called a (right) semientwining map if it satisfies the following conditions for all , , (where we use a Sweedler-like summation notation ):
If is also an -algebra, and a semientwining map satisfies additionally
then the semientwining map is called an algebra factorization (in the sense of ).
If is a coalgebra and satisfies then is called a (left-left) entwining map .
Remark 4. Let . The following are examples of semientwining structures. Note that they do not have natural algebra factorization structures in general. (1)Let be an -algebra, then the -linear map , is a semientwining map. Notice that is a Yang-Baxter operator (according to ).(2)Let be an -algebra, then the -linear map , is a semientwining map. Notice that is a Yang-Baxter operator related to Lie algebras (see, e.g., ).(3)Let be an -algebra, and let be a right -module. Then the -linear map , is a semientwining map.
The proof of the next lemma is direct; the second statement is a well-known result.
Lemma 5. If is a semientwining map, then(i) becomes a right -module with the operation ;(ii)moreover, if is an algebra, we can define a bilinear operation and is an associative and unital multiplication on if and only if is an algebra factorization.
Remark 6. Some authors call the above map a twisting map; see, for example, , where a unifying framework for various twisted algebras is provided.
Remark 7. Suppose that is a right -comodule algebra (where is a bialgebra), and is a right -module. Then is a semientwining map. Moreover, if is an -module algebra, then thus defined is an algebra factorization. Finally, if is an -module coalgebra, then is an entwining map, and is called a Doi-Koppinen structure (see ).
Remark 8. Let be an -algebra. We define the category of semientwining structures over , whose objects are triples , and morphisms are -linear maps satisfying the relation . Then, there exist the following functors. (1): Mod SemiEntwining Str . , where ;(2): SemiEntwining Str Mod . , where is a right -module with the operation .
These two functors do not form an equivalence of categories in general, because and .
Theorem 9. If is a semientwining map, and is bialgebra, then(1) is an -bimodule with the following actions: (2) is an algebra with the unit and the product and a right -comodule with the coaction .(3)If has a bilateral integral (i.e.,) which is a group-like element (i.e.,), then is an -comodule algebra with the coaction
Proof. Follows from the linearity of and .
Follows from the previous statement and from direct computations as follows: maps to either (if we apply the comultiplication of the algebra), or to ) (if we apply the coaction).
We observe that the two outputs are equal.
Is a generalisation of and is left to the reader.
Similarly we have the dual notion as follows.
Definition 10 (cosemientwining structures). Let be an -coalgebra, and let be an -module. A -linear map is called a cosemientwining map if it satisfies the following conditions for all , , (where we use a Sweedler-like summation notation ):
If is also a coalgebra, and satisfies additionally
then is called a coalgebra factorization.
If, on the other hand, is an algebra, and satisfies additionally then is called a (right-right) entwining map.
The next result is dual to Lemma 5.
Lemma 11. Suppose that is a cosemientwining map, and is a coalgebra. Define a map Then makes a coalgebra if and only if is a coalgebra factorization.
Proof. For to be a coalgebra it must satisfy the counit property, that is, and the coassociativity property. To check a counit property note that for all and :
Now, if , then applying to both sides of this equation yields . Similarly, we prove the other half of the counit property. Conversely, implies the counit property.
Using the fact that is a cosemientwining map, it is easy to prove that the coassociativity implies that for all and Applying to the third leg and using the fact that is a cosemientwining map yields We leave the rest of the proof to the reader.
Remark 12. Suppose that is a right -comodule coalgebra (where is a bialgebra), and is a right -module. Then is a cosemientwining map. Furthermore, if is an -module coalgebra, then is a coalgebra factorization. Otherwise, if is an -module algebra, then is a left-left entwining map. Moreover, in this last case, is called an alternative Doi-Koppinen structure.
Let , be any -modules. Any can be viewed as the map Also any tensor can be considered as a map . Finally, if is finitely generated and projective, then . For any , an -module map defines a map We define a dual of with respect to the -part as , where is defined by Similarly, one defines a dual of with respect to the -part.
The next lemma is a standard result.
Lemma 13. Suppose that is a finitely generated projective -coalgebra, and is a dual basis. Let be a cosemientwining map. Then is a semientwining map for the convolution algebra .
Definition 14 (semientwined modules and comodules). Let be an algebra, and let be a vector space. Suppose that is a semientwining map, and a right module.(1)Let be a right measuring, such that for all , , , Then is called a semientwined module. (2)Let , be a right comeasuring, such that for all , , Then is called a semientwined comodule.
Remark 15. The following are examples of semientwining modules related to Remark 4: (1)let be an -algebra, let be a right module, , , and the right measuring the regular action of on ; (2)let be an -algebra, let be a right module, , , and the right measuring the regular action of on .
Remark 16. The following are examples of semientwining comodules related to Remark 4: (1)let be an -algebra, let be a right module, , , and the right comeasuring ; (2)let be an -algebra, let be a right module, , , and the right comeasuring .
Definition 17 (cosemientwined modules and comodules). Let be a coalgebra, and let be a vector space. Suppose that is a cosemientwining map, and a left -comodule, with a coaction , . (1)Let be a left measuring, such that for all , , Then is called a cosemientwined module. (2)Let , be a left comeasuring, such that for all , Then is called a cosemientwined comodule.
Note that if is a coalgebra, and is an entwining map, then a semientwined module is an entwined module.
The following result is standard, but we provide a partial proof for completeness.
Lemma 18. Suppose that is an algebra factorization, and is a semientwined module, such that the measuring is an action. Then is a right -module, with an algebra structure on as in Lemma 5, and action on given by . Conversely, any right module is a semientwined module with and actions given by and , respectively.
Proof. It is enough to verify that the definition of action agrees with the algebra relations, that is, that Both sides of the above equation equal -left one because of algebra relations, and the right one because is a semientwined module. We prove similarly the rest of the lemma.
3.1. Intertwining Operators
We give a brief introduction to the intertwining operators below.
Let be an -algebra. Given two algebra representations, say and , we define an intertwining operator to be a linear operator, such that .
With this definition we can define the category of finite-dimensional representations of , in which the morphisms are intertwining operators (see ).
The following theorem provides a connection between semientwining structures and intertwining operators.
Theorem 19. Let be an -algebra, let be an -module, and let be a semientwining map. Then, the following statements are true:(i) is a right -module in a trivial way, with the right action, . (ii) is a right -module in the following way: , .(iii)With the above actions, is an intertwining operator (i.e., satisfies the relation ).
3.2. Braided Algebras
Many algebras obtained by quantization are commutative braided algebras, and all super-commutative algebras are automatically commutative braided algebras (see ).
Definition 20. An algebra for which there exists a Yang-Baxter operator such that , , , and for all is called a braided algebra.
Moreover, if for all , we call a commutative braided algebra or an -commutative algebra (see ).
Definition 21. Given braided algebras and , we say that is a braided algebra morphism if it is a morphism of algebras and (see ).
Theorem 22. (i) Any algebra becomes a commutative braided algebra with .
(ii) If and are two braided algebras as in (i), and is an algebra morphism, then it is also a braided algebra morphism.
(iii) If is a derivation (i.e., and ), then there exists a morphism of braided algebras , , where and .
Proof. (i) Notice that is a self-inverse Yang-Baxter operator which was studied in [16, 21]. , (directly) (from Remark 4 (i) with ) = + + −
(ii) This follows from Proposition 3.1 of . Also, refer to .
(iii) The proof is direct and is left to the reader.
Remark 23. In the above example ; so, the above algebra is “strong.” All sorts of noncommutative analogs of manifolds are commutative braided algebras: quantum groups, noncommutative tori, quantum vector spaces, the Weyl and Clifford algebras, certain universal enveloping algebras, super-manifolds, and so forth. It seems that the ones with direct relevance to quantum theory in 4 dimensions are “strong,” while the nonstrong ones, like quantum groups, are primarily relevant to 2- and 3-dimensional physics (see ).
3.3. Liftings of Functors
The semientwining structures can be understood as liftings of functors from one category to another. This goes back as far back as . This situation is reviewed in : the semientwining case is dealt with in general in item 3.3 (which is transferred from ); how this general case is translated to our situation is clear from the discussion in item 5.8 of . This is also presented in Section 3.1 of , where the axioms of semientwining structures are given by formula (3.1).
We give a general definition of liftings of functors. is a lifting of if the following diagram commutes (28) where and are forgetful functors.
We now present examples of liftings of functors related to semientwining structures.
Theorem 24. Let be an -algebra, and let be an -module. The functor can be lifted from the category of -modules to the category of right -modules there exists a -linear map which is a semientwining map.
Proof. Assume that there exists a semientwining , then lifts to a functor which associates to a right -module the -module with a right action given by It remains to check that for any right -module function , the map is a right -module map as follows: On the other hand, suppose that lifts to a functor in the category of right -modules. In particular, it follows that is a right -module. Define the linear map by the formula We shall prove that this is a semientwining map. Indeed, by definition we have Any element defines a right -module map It follows that for any , we have from the -linearity of as follows: Hence
Remark 25. Let be an -algebra, and let be an -module. Using our terminology (given in Remark 8) and the results of , we conclude that the category of semientwining structures over is isomorphic to the category of lifting of functors from the category of -modules to the category of right -modules.
Remark 26. We now give a more general definition than that given in Remark 8.
We define the category of semientwining structures, whose objects are triples , and morphisms are pairs where is an -linear map, is an algebra morphism, and they satisfy the relation .
In a dual manner, let us define the category of cosemientwining structures, whose objects are triples , and morphisms are pairs where is an -linear map, is a coalgebra morphism, and they satisfy the relation .
The duality functor from the category of coalgebras to the category of algebras can be lifted to a functor from the category of cosemientwining structures to the category of semientwining structures (by Lemma 13).
This fact is described in the following diagram: (36)
Remark 27. A braided coalgebra is a structure dual to Definition 20 (see, e.g. ).
The duality between finite-dimensional algebras and finite-dimensional coalgebras can be lifted to a duality between the categories of finite-dimensional-braided algebras and finite- dimensional braided coalgebras. This fact is described in the following diagram: (37)
3.4. Tambara Bialgebras
Definition 28 (Tambara bialgebra ). Let be a finitely generated and projective -algebra (which implies that is a coalgebra), and let , , be a dual basis of . Let be an ideal generated by elements for all , . Then is called a Tambara bialgebra. Denoting by the class of in , the comultiplication and counit is given by is a right -comodule algebra with coaction
Theorem 29 (see ). Suppose that is a finitely generated projective -algebra, and is an -module. Then semientwining structures are in one-to-one correspondence with right -module structures on . Similarly, if is an algebra, then algebra factorizations are in one to one correspondence with right -module algebra structures on . Finally if is a coalgebra, then entwining structures are in one to one correspondence with right -module coalgebra structures on . Explicitly, given right -module structure on , we define (7). Conversely, given a semientwining , we define a right module action on by
Remark 30. Let . The examples of semientwining structures presented in Remark 4 generate the following structures:(1)a right module action on by
(2)a right module action on by
(3)a right module action on , for any right -module , by
Let be a finitely generated and projective coalgebra. Let , , be a dual basis of . Note that where ) is an ideal generated by elements for all , , , with explicit coaction and counit given by
Theorem 31 (see ). Suppose that is a finitely generated projective -coalgebra, and is an -module. Then cosemientwining structures are in one-to-one correspondence with right module structures on . Similarly if is a coalgebra, then coalgebra factorizations are in one to one correspondence with -module coalgebra structures on . Finally, if is an algebra, then (right-right) entwining structures are in one to one correspondence with right -module algebra structures on . Explicitly, given right -module structures on , we define (18). Conversely, given a cosemientwining , we define a right -module structures on by
3.5. Yang-Baxter Systems
From now on we work over a field . It is convenient to introduce the constant Yang-Baxter commutator of the linear maps In this notation, the quantum Yang-Baxter equation reads .
Definition 32 (Yang-Baxter systems of type I). A system of linear maps of vector spaces , , is called a system (or a Yang-Baxter system of type I) if A system of linear maps , satisfying (49) is called a semi Yang-Baxter system. One can associate a system to a semi Yang-Baxter system by setting .
Remark 33. From a Yang-Baxter system of type I, one can construct a Yang-Baxter operator on , provided that the map is invertible (see ).
Let be an algebra, and the map for some arbitrary , (see ). Then, .
The following is an enhanced version of Theorem 2.3 of .
Theorem 34 (see ). Let be an algebra, let be a vector space, and .
Let , and let be a linear map, such that , for all .(i)Then is a semi Yang-Baxter system if and only if is a semientwining map.(ii)Similarly, if is an algebra, , and , for all , then is a Yang-Baxter system of type I if and only if is an algebra factorization.
Definition 35 (Yang-Baxter systems of type II). A system of linear maps of vector spaces is called a Yang-Baxter system of type II if where (and is the twist map).
Remark 36. Yang-Baxter systems of type II are related to the algebras considered in , which include (algebras of functions on) quantum groups, quantum super-groups, braided groups, quantized braided groups, reflection algebras, and others.
The following theorems present solutions for the Yang-Baxter systems.
Theorem 37 (see ). Let be a commutative algebra, and . Then, , and is a Yang-Baxter system of type II.
Theorem 38. Let , , in the above theorem. It turns out that , , is also a Yang-Baxter system of type I.
Proof. First, let us observe that the result holds even for a noncommutative algebra. One way to prove the theorem is by direct computations.
Alternatively, one can observe that is an algebra factorization, and apply Remark 2.4 of .
Also, refer to Theorem 5.2 of .
Theorem 40. Let be an algebra; ; semientwining maps; , , , . If , then is a Yang-Baxter system of type II.
Proof. Use Theorem 34 (i) to check the first-four equations. Then, observe that . The last-four equations then follow.
Theorem 41. Let be an algebra, and a semientwining map.
Then, there exists a semientwining map , such that if and only if , viewed as , is an algebra factorization.
Proof. Assume that there exists a semientwining map . Denote , for all , that is, . Also denote by the multiplication in , that is, for all , . Then we must check conditions (4). For all ,
Similarly one can prove the converse.
Remark 42 (example of algebra factorization for Theorem 41). We consider the algebra , where is a scalar. Then has the basis , where is the image of in the factor ring, so .
If is a scalar, then , defined as follows is an algebra factorization.
Notice that if , then is the same algebra factorization with (53).
Theorem 43. Let be an algebra, let and be vector spaces, , a semientwining, and let be an semientwined module with the right measuring . We consider the maps as follows: Then, the following equation holds
Proof. The proof follows by direct computations.
Remark 44. The relation from the above theorem is related to Section 3.6 of .
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