Semi-entwining structures and their applications

Semi-entwining 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, etc. While for entwining structures one can associate corings, for semi-entwining structures one can associate comodule algebra structures where the algebra involved is a bialgebra satisfying certain properties.


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 [15] 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 [7] as generalized symmetries of noncommutative principal bundles, and provide a unifying framework for various Hopftype modules. They are related to the so called mixed distributive laws introduced in [4].
The Yang-Baxter systems emerged as spectral-parameter independent generalization of the quantum Yang-Baxter equation related to non-ultra-local integrable systems [12,13]. Interesting links between the entwining structures and Yang-Baxter systems have been established in [8] and [5]. Both topics have been a focus of recent research (see, for example, [18,19,20,24,9,22]).
In this paper, we propose the concepts of semi-entwining structures and cosemientwining structures within a generic framework incorporating results of other authors alongside ours. The semi-entwining structures are some kind of entwining structures between an algebra and a module which obey only one half of their axioms, while cosemi-entwining 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 semi-entwining 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 semi-entwining structures one can associate comodule algebra structures provided the algebra involved is a bialgebra with certain properties (see Theorem 2.7). 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 type I and II).
The main results of our paper are Theorems 3.1, 3.4, 3.6, 3.22, and 3.23. Theorems 3.11 and 3.13 are mentioned in the context of stating some of our results. Theorem 3.16 is used to prove Theorem 3.22, while Theorem 3.19 is related to Theorem 3.20.
Unless otherwise stated, we work over a commutative ring R. Unadorned tensor products mean tensor products over R.
For any R-module V , T (V ) denotes tensor algebra of V . In Subsection 3.5, we work over a field K. For V an R-module, we denote by I : V → V the identity map. For any R-modules V and W we denote by We use the following notations: (1) 2. Semi-entwining structures and related structures Definition 2.1. [Semi-entwining structures] Let A be an R-algebra, and let B be an R-module, then the R-linear map ψ : B ⊗ A → A ⊗ B is called a (right) semientwining map if it satisfies the following conditions for all a, a ′ ∈ A, b ∈ B (where we use a Sweedler-like summation notation ψ(b ⊗ a) = a α ⊗ b α ): If B is also an R-algebra, and a semi-entwining map satisfies additionally then the semi-entwining map is called an algebra factorization (in the sense of [6]).
If B is a coalgebra and satisfies then ψ is called a (left-left) entwining map [7].
The following are examples of semi-entwining structures. Note that they do not have natural algebra factorization structures in general.
(1) Let A be an R-algebra, then the R-linear map γ q : Notice that γ q is a Yang-Baxter operator (according to [10]). (2) Let A be an R-algebra, then the R-linear map η q : Notice that η q is a Yang-Baxter operator related to Lie algebras (see, e.g., [16]).
(3) Let A be an R-algebra and let M be a right A-module. Then the R-linear map The proof of the next lemma is direct; the second statement is a well-known result.
is an algebra, we can define a bilinear operation and · is an associative and unital multiplication on A⊗B if and only if ψ is an algebra factorization.
Remark 2.4. Some authors call the above map ψ a twisting map; see, for example, [21], where a unifying framework for various twisted algebras is provided.
is a semi-entwining map. Moreover, if B is an H-module algebra, then ψ H thus defined is an algebra factorization. Finally, if B is an H-module coalgebra, then ψ is an entwining map, and (A, H, B) is called a Doi-Koppinen structure (see [22]).
Remark 2.6. Let A be an R-algebra. We define the category of semi-entwining structures over A, whose objects are triples (B, A, φ), and morphisms f : . Then, there exist the following functors.
These two functors do not form an equivalence of categories in general, because Theorem 2.7. If ψ : B ⊗ A → A ⊗ B is a semi-entwining map and A is bialgebra, then: (1) B is an A-bimodule with the following actions: (2) B ⊕ A is an algebra with the unit (0, 1) and the product (b, a) (b ′ , a ′ ) = (b * a ′ + a • b ′ , aa ′ ), and a right A-comodule with the coaction b ⊕ a → b ⊗ 1 + ( a 1 ⊗ a 2 ).
(2) follows from the previous statement and from direct computations as follows: ) (if we apply the comultiplication of the algebra), or to We observe that the two outputs are equal.
(3) is a generalisation of (2), and is left to the reader.
Similarly we have the dual notion as follows.
If D is also a coalgebra and ψ satisfies additionally then ψ is called a coalgebra factorization. If, on the other hand, D is an algebra, and ψ satisfies additionally The next result is dual to Lemma 2.3. Lemma 2.9. Suppose that ψ : D ⊗ C → C ⊗ D is a cosemi-entwining map and D is a coalgebra. Define a map Then ∆ makes D ⊗ C a coalgebra if and only if ψ is a coalgebra factorization.
Proof. For D ⊗ C to be a coalgebra it must satisfy the counit property, i.e., (ε D⊗C ⊗ id) • ∆ D⊗C = (id ⊗ ε D⊗C ) • ∆ D⊗C = id and the coassociativity property. To check a counit property note that for all d ∈ D and c ∈ C: , then applying ε ⊗ id to both sides of this equation yields cε(d) = c α ε(d α ). Similarly, we prove the other half of the counit property. Conversely, cε(d) = c α ε(d α ) implies the counit property.
Using the fact that ψ is a cosemi-entwining map it is easy to prove that the coassociativity implies that for all c ∈ C and d ∈ D Applying ε to the third leg and using the fact that ψ is a cosemi-entwining map yields We leave the rest of the proof to the reader.
is a cosemi-entwining map. Furthermore, if D is an H-module coalgebra, then ψ is a coalgebra factorization. Otherwise, if D is an H-module algebra, then ψ is a left-left entwining map. Moreover, in this last case, (C, H, D) is called an alternative Doi-Koppinen structure.
Let X, Y be any R-modules. Any x * ∈ X * can be viewed as the map We define a dual of Ψ * X : Y ⊗X * → X * ⊗Y with respect to the X-part as Ψ * The next lemma is a standard result.
Lemma 2.11. Suppose that C is a finitely generated projective R-coalgebra, and Explicitly Definition 2.12.
[Semi-entwined modules and comodules] Let A be an algebra, and let V be a vector space. Suppose that ψ : Then M is called a (A, V, ψ)-semi-entwined module.
Remark 2.13. The following are examples of semi-entwining modules related to Remark 2.2: (1) let A be an R-algebra, let M be a right A module, V = A, ψ = γ q , and the right measuring the regular action of A on M; (2) let A be an R-algebra, let M be a right A module, V = A, ψ = η 1 , and the right measuring the regular action of A on M.
Remark 2.14. The following are examples of semi-entwining comodules related to Remark 2.2: (1) let A be an R-algebra, let M be a right A module, V = A, ψ = γ 1 , and the right comeasuring ρ(m) = m ⊗ 1; (2) let A be an R-algebra, let M be a right A module, V = A, ψ = η q , and the right comeasuring ρ(m) = m ⊗ 1.

Definition 2.15. [Cosemi-entwined modules and comodules]
Let C be a coalgebra, and V a vector space. Suppose that ψ : Then M is called a (C, V, ψ)-cosemi-entwined module.
Note that if V is a coalgebra and ψ : V ⊗ A → A ⊗ V is an entwining map, then a semi-entwined module M is an entwined module.
The following result is standard, but we provide a partial proof for completeness. Proof. It is enough to verify that the definition of A⊗B action agrees with the algebra relations, i.e., that m((1 ⊗ b)(a ⊗ 1)) = (m(1 ⊗ b))(a ⊗ b) Both sides of the above equation equal ma α ⊳ b α -left one because of algebra relations, and the right one because M is a (A, B, ψ) semi-entwined module. We prove similarly the rest of the lemma.

Intertwining Operators.
We give a brief introduction to the intertwining operators below.
Let A be an R-algebra. Given two algebra representations, say ρ : With this definition we can define the category of finite dimensional representations of A, in which the morphisms are intertwining operators (see [3]).
The following theorem provides a connection between semi-entwining structures and intertwining operators.
Theorem 3.1. Let A be an R-algebra, let B be an R-module, and let ψ : B ⊗ A → A ⊗ B be a semi-entwining map. Then, the following statements are true: (i) B⊗A is a right A-module in a trivial way, with the right action ρ : Proof. The proof of (i) is direct and (ii) follows from Lemma 2.3(i). The relation ψ • ρ = ρ ′ • (ψ ⊗ I) is equivalent to the second relation of (3).

Braided
Algebras. Many algebras obtained by quantization are commutative braided algebras, and all super-commutative algebras are automatically commutative braided algebras (see [1]).
(iii) The proof is direct and is left to the reader.
Remark 3.5. In the above example ψ • ψ = I ⊗ I; so, the above algebra is "strong". All sorts of non-commutative analogs of manifolds are commutative braided algebras: quantum groups, non-commutative 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 non-strong ones, like quantum groups, are primarily relevant to 2-and 3dimensional physics (see [1]).

Liftings of Functors.
The semi-entwining structures can be understood as liftings of functors from one category to another. This goes back as far back as [14]. This situation is reviewed in [24]: the semi-entwining case is dealt with in general in item 3.3 (which is transfered from [14]); how this general case is translated to our situation is clear from the discussion in item 5.8 of [24]. This is also presented in subsection 3.1 of [25], where the axioms of semi-entwining structures are given by formula (3.1). We give a general definition of liftings of functors. F is a lifting of G if the following diagram commutes: where U and U ′ are forgetful functors. We now present examples of liftings of functors related to semi-entwining structures.

It remains to check that for any right
On the other hand, suppose that − ⊗ B lifts to a functor in the category of right A-modules. In particular, it follows that A ⊗ B is a right A-module. Define the linear map Ψ : We shall prove that this is a semi-entwining map. Indeed, by definition we have Any element a ∈ A defines a right A-module map It follows that for any a ′ ∈ A we have from the A-linearity of f ⊗ id: Let A be an R-algebra and let B be an R-module. Using our terminology (given in Remark 2.6) and the results of [25], we conclude that the category of semientwining structures over A is isomorphic to the category of lifting of functors from the category of R-modules to the category of right A-modules.
Remark 3.8. We now give a more general definition than that given in Remark 2.6.
We define the category of semi-entwining structures, whose objects are triples (B, A, φ), and morphisms are pairs (f, g) : (B, A, φ) → (B ′ , A ′ , φ ′ ) where f : B → B ′ is an R-linear map, g : A → A ′ is an algebra morphism, and they satisfy the relation In a dual manner, let us define the category of cosemi-entwining structures, whose objects are triples (D, C, φ), and morphisms are pairs (f, g) : (D, C, φ) → (D ′ , C ′ , φ ′ ) where f : D → D ′ is an R-linear map, g : C → C ′ 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 cosemi-entwining structures to the category of semi-entwining structures (by Lemma 2.11).
This fact is described in the following diagram.
Cosemi − entw str A braided coalgebra is a structure dual to Definition 3.2 (see, e.g. [11]). 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.

Tambara bialgebras.
Definition 3.10. [Tambara Bialgebra ([23])] Let A be a finitely generated and projective R-algebra (which implies that A * is a coalgebra), and let a i , a * i , i = 1, . . . , N be a dual basis of A. Let I ⊂ T (A * ⊗ A) be an ideal generated by elements Denoting by [a * ⊗ a] the class of a ⊗ a * in H(A), the comultiplication ∆ and counit ε is given by A is a right H(A)-comodule algebra with coaction Let C be a finitely generated and projective R-coalgebra. Let c i , c * i , i = 1, . . . , N be a dual basis of C. Note that H(C * ) cop = T (C * ⊗ C)/I ′ where I ′ ⊂ T (C * ⊗ C) is an ideal generated by elements for all c * , d * ∈ C * , c ∈ C, with explicit coaction and counit given by 3.5. Yang-Baxter Systems. From now on we work over a field K. It is convenient to introduce the constant Yang-Baxter commutator of the linear maps In this notation, the quantum Yang-Baxter equation reads [R, R, R] = 0.
[Yang-Baxter systems of type I] A system of linear maps of vector spaces W : A system of linear maps W, X satisfying equations (8a) is called a semi Yang-Baxter system. One can associate a WXZ system to a semi Yang-Baxter system by setting Z = I ⊗ I. Remark 3.15. From a Yang-Baxter system of type I, one can construct a Yang-Baxter operator on (V ⊕ V ) ⊗ (V ⊕ V ), provided that the map X is invertible (see [8]).
Let A be an algebra, and the map for some arbitrary s, r ∈ K (see [10]). Then, [W, W, W ] = 0.
The following is an enhanced version of Theorem 2.3 of [8].
Theorem 3.16. (see [8]) Let A be an algebra, let B be a vector space, and p, q, s, r ∈ K.
Let W = R A r,s , and let X : i) Then W, X is a semi Yang-Baxter system if and only if ψ = X • τ B,A is a semi-entwining map.
ii) Similarly, if B is an algebra, Z = R B p,q , and X(a⊗1 B ) = a⊗1 B for all a ∈ A, then W, X, Z is a Yang-Baxter system of type I if and only if ψ is an algebra factorization. Remark 3.18. Yang-Baxter systems of type II are related to the algebras considered in [12], which include (algebras of functions on) quantum groups, quantum supergroups, braided groups, quantized braided groups, reflection algebras and others.
The following theorems present solutions for the Yang-Baxter systems.
Theorem 3.19. (see [19]) Let A be a commutative algebra and λ, λ ′ ∈ K. Then, Theorem 3.20. Let W = A, X = B = C, Z = D in the above theorem. It turns out that W, X, Z is also a Yang-Baxter system of type I.

Proof
First, let us observe that the result holds even for A a non-commutative algebra. One way to prove the theorem is by direct computations. Alternatively, one can observe that (10) ψ(a ⊗ b) = 1 ⊗ ab + ab ⊗ 1 − a ⊗ b is an algebra factorization, and apply Remark 2.4 of [8]. Also, refer to Theorem 5.2 of [18].
Remark 3.21. One can combine the proof of the Theorem 3.20 with Remark 2.4 and Proposition 2.9 of [8], to obtain a large class of Yang-Baxter operators defined on V ⊗ V , where V = A ⊕ A. See also Remark 3.15.
Theorem 3.23. Let A be an algebra and ψ : A ⊗ A → A ⊗ A a semi-entwining map.
Then, there exists a semi-entwining map ψ ′ : A⊗A → A⊗A such that ψ ′ = τ •ψ •τ if and only if ψ, viewed as ψ : A op ⊗ A → A ⊗ A op , is an algebra factorization.
Proof. Assume that there exists a semi-entwining map ψ ′ = τ • ψ • τ . Denote ψ ′ (a ⊗ b) = b α ′ ⊗ a α ′ , for all a, b ∈ A, i.e., a α ⊗ b α = a α ′ ⊗ b α ′ . Also denote by · op the multiplication in A op , i.e., for all a, b ∈ A, a · op b ≡ ba. Then we must check conditions of Definition 2.1. For all a, b, c ∈ A, Similarly one can prove the converse.

Remark 3.24. [Example of algebra factorization for Theorem 3.23]
We consider the algebra A = A op = K[X] (X 2 −p) , where p is a scalar. Then A has the basis {1, x}, where x is the image of X in the factor ring; so, x 2 = p.
If q is a scalar, then ψ : A op ⊗ A → A ⊗ A op , defined as follows, is an algebra factorization. Notice that if q = 2p, then ψ is the same algebra factorization with (10). [ζ, η, X] = 0