Non-commutative geometry is a branch of mathematics concerned with geometric approach to non-commutative algebras, and with constructions of spaces which are locally presented by non-commutative algebras of functions. Its main motivation is to extend the commutative duality between spaces and functions to the non-commutative setting.

More specifically, in topology, compact Hausdorff topological spaces can be reconstructed from the Banach algebra of functions on the space. The Pontryagin duality theorem refers to the duality between the category of compact Hausdorff Abelian groups and the category of discrete Abelian groups. The Pontryagin–van Kampen duality theorem extends this duality to all locally compact Hausdorff Abelian topological groups by including the categories of compact Hausdorff Abelian groups and discrete Abelian groups into the category of locally compact Hausdorff Abelian topological groups (see [1]). This can be illustrated by the following diagram:

Taking the Pontryagin–van Kampen duality theorem as a model, an extension for the duality between finite dimensional algebras and coalgebras to the category of finite dimensional Yang–Baxter structures was constructed in [2]. The resulting duality theorem can be illustrated by the following diagram:

Our motivation in this paper is to extend the above duality to the non-commutative setting.

In Section 2, we present in a new fashion the duality between right finitely generated projective corings and ring extensions (compare with [3]).

In Section 3, we define the category of (right finitely generated projective) generalized Yang–Baxter structures. We construct full and faithful embeddings from the categories of ring extensions and corings to the category of generalized Yang–Baxter structures. We show that taking the right dual is a duality functor in the category of right finitely generated projective generalized Yang–Baxter structures. Then we conclude that the duality between right finitely generated projective corings and ring extensions can be lifted up to the category of right finitely generated projective generalized Yang–Baxter structures.

There are some more comments to be made.

Throughout this paper 𝕂 is a commutative ring, and all 𝕂-modules M are such that for all m∈ M, 2m = 0 implies m = 0.

Let A, B, C, etc. be algebras over ground commutative ring 𝕂. Unadorned tensor product will denote the tensor product over 𝕂. For modules M in _AB, symbols M* , *M, *M* denote right dual, left dual and bidual of M, and _AB (M, N) denotes the 𝕂-module of (A, B)-bimodule maps M → N. In what follows we shall concentrate on right dual of M but similar observations can be made for the left dual as well.

For all ϕ∈_AB (M, N), let ϕ* : N* → M* denote the right adjoint of ϕ i.e., ϕ* (g)(m) := g ◦ ϕ (m).

We denote by (·)^op : A → A^op the canonical anti-algebra isomorphism from the algebra A into its opposite A^op (which is the identity on the underlying 𝕂-modules), i.e., a = a^op as module elements and (aa´)^op = a´^opa^op for all a, a´∈ A.

The following facts are well known, but we recall them to set up the notation:

(i) If M∈_AB then M* ∈ _A^op_B^op with (a^opfb^op)(m) = bf(am).

Assume that M∈_AB is also finitely generated projective as a right B-module, i.e., there exists a dual basis , such that for any m∈ M, . Then

(ii) The mapping κ_M: M → M**, κ_M (m)(f) = f(m)^op is an isomorphism in _AB, with the inverse . In fact κ is a natural morphism between identity functor in _AB and the functor ()^** : _AB → _AB.

(iii) If N∈_BC then κ_M,N : M* ⊗ _B^op N* → (M ⊗ _B N)*, given by κ_M,N (f ⊗ _B^op g)(m ⊗ n) = g(f(m)n), is an isomorphism in _A^op_C^op with the inverse

(iv) Let M∈ _AB, N∈ _BC , P ∈_CD, where A, B, C, D are algebras. Then the following diagram is commutative:

(v) Let M∈ _AB be finitely generated projective as B-module, with dual basis , i∈ I, and let N∈_BC be finitely generated projective as a C-module with dual basis , , i∈ J. Then M ⊗ _B N∈_AC is finitely generated projective as a C-module with a dual basis

The following terminology and theorems concerning corings and ring extensions are needed in this paper. For a review on coalgebras see: [5,6,7]. For a review on corings see [3].

Definition 2.1 C∈_BB is called a B-coring if there exist morphisms Δ^C, ε^C ∈_BB, Δ^C : C→ C ⊗ _B C, ε^C : C→ B such that

In the sequel we shall use Sweedler’s notation Δ^C(c) = c₍₁₎ ⊗ _B c₍₂₎. Given B-corings C and D, a map ϕ ∈_BB (C, D) is called a morphism of B-corings if (ϕ ⊗ _Bϕ) ◦ Δ^C = Δ^D ◦ ϕ and ε^D ◦ ϕ = ε^C. The category of B-corings is denoted by Crg_B.

Definition 2.2 Ring is called an extension of a ring B if there exists an injective unital ring morphism : B →. Observe that ∈_BB by . Given ring extensions : B → and : B →P, a ring morphism α : →P is called a morphism of ring extensions if α ◦ = or, equivalently, if α ∈ _BB (, P). The category of ring extensions of B is denoted by Rge_B.

The full subcategory of Crg_B (resp. Rge_B) consisting of those B-corings (resp. ring extensions of B) that are finitely generated projective as right B-modules is denoted by r.f.g.pCrg_B (resp. r.f.g.pRge_B).

Lemma 2.3 (i) If C∈ Crg_B then C* ∈_B^op_B^opis a ring extension of B^op with multiplication

unit 1_C* := ε^C and embedding map

(ii) If ϕ : C→D is any coring morphism then ϕ*: D* →C* is a ring extension morphism.

(iii) If ∈ r.f.g.pRge_B then * is a B^op-coring with comultiplication and counit

where is a (finite) dual basis of .

(iv) If ϕ : →S is a morphism of right finitely generated projective ring extensions of B, then : S* →* is a morphism of B^op-corings.

(v) Functor ()**: r.f.g.pRge_B → r.f.g.pRge_B is equivalent to the identity functor on r.f.g.pRge_B. For all ∈ r.f.g.pRge_B, κ_R : →** is a ring extension isomorphism facilitating this equivalence.

(vi) Functor ()**: r.f.g.pCrg_B → r.f.g.pCrg_B is equivalent to the identity functor on r.f.g.pCrg_B. For all C∈ r.f.g.pCrg_B, κ_C : C→C** is a B-coring isomorphism facilitating this equivalence.

Proof. The statements (i) and (ii) are contained in Proposition 3.2 [8], while (iii) and (v) are rephrasings of Theorem 3.7 [8] (cf. [3], 17.8–17.13)

(iv) Consider any ring extension morphism ϕ : →S. Let be any finite dual basis of , and let be any finite dual basis of S. For all s∈S*,

and

Hence ϕ* is a coring map.

(vi) It is enough to prove that κ_C, is a coring map for any C∈ r.f.g.pCrg_B. Let C be a B-coring, and let , be any finite dual basis of C. Observe that is a dual basis of C*. Indeed, for any g∈C*,

Hence, for all c∈C

and

Corollary 2.4 ()* is a duality functor between r.f.g.pRge_B and r.f.g.pCrg_B^op

Our aim in this section is to extend the duality between right finitely generated projective ring extensions and corings to the category of right finitely generated projective generalized Yang–Baxter structures.

We use the following terminology concerning the Yang–Baxter equation. Some references on this topic are: [9,10,11], etc.

Let B be a 𝕂-algebra. Given a (B, B)-bimodule V and a (B, B)-bilinear map R : V ⊗ _B V → V ⊗ _B V we write R¹² = R ⊗ _B id, R²³ = id ⊗ _B R : V ⊗ _B V ⊗ _B V → V ⊗ _B V ⊗ _B V where id : V → V is the identity map.

Definition 3.1 An invertible (B, B)-linear map R : V ⊗ _B V → V ⊗ _B V is called a generalized Yang–Baxter operator (or simply a generalised YB operator ) if it satisfies the equation

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Definition 3.2 For an algebra B, we define the category YB str_B whose objects are 4-tuples (V, φ, e, ε), where

(i) V is a (B, B)-bimodule;

(ii) φ : V ⊗ _B V → V ⊗ _B V is a generalized YB operator;

(iii) e∈ V such that for all b∈ B, eb = be, and for all x∈ V , φ(x ⊗ e) = e ⊗ _B x, φ(e ⊗ _B x) = x ⊗ _B e;

(iv) ε : V → B is a (B, B)-bimodule map, such that (id ⊗ _B ε) ◦φ = ε ⊗ _B id, (ε ⊗ _B id) ◦φ = id ⊗ _B ε.

A morphism f :(V, φ, e, ε) → (V’, φ’,e’,ε’) in the category YB str_B is a (B, B)-bilinear map f : V → V’ such that:

(v) (f ⊗ _B f) ◦φ = φ´◦ (f ⊗ _B f),

(vi) f(e) = e´,

(vii) ε^´ ◦ f = ε.

Composition of morphisms is defined as the standard composition of B-linear maps. A full subcategory of YB str_B consisting of all such (V, φ, e, ε) for which V is finitely generated projective as a right B-module is defined by r.f.g.pYBstr_B.

Remark 3.3 Let R : V ⊗ _B V → V ⊗ _B V be a generalised YB operator . Then (V, R, 0, 0) is an object in the category YB str_B.

Theorem 3.4 (i) There exists a functor:

Any ring extension map f is simply mapped into a (B, B) bimodule map.

(ii) F is a full and faithful embedding.

Proof. i) The proof that φ_R is a generalised YB operator is left to the reader (cf. Proposition 2.1 from [12], ). Furthermore Hence (R, φ_R, 1_R, 0) is an object in the category YB str_B.

Let f : →S be a morphism of ring extensions. Then f(1_R)= 1_S and 0 ◦ f = 0. Moreover

Hence is a morphism in the category YB str_B.

(ii) If F = F S, for some , S∈ Rge_B, then obviously = S as (B, B)-bimodules, 1_S = 1_R, and the only thing which can differ is the multiplication. Denote by · the multiplication in , and by ◦ the multiplication in S. Then, as φ_R = φ_S , for all r, r´∈,

hence

Multiplying tensor factors on both sides of this equation (whether using multiplication in or S is irrelevant) yields 2(r · r´− r ◦ r´) = 0, hence r · r = r ◦ r´, and so = S as algebras. Therefore F is an embedding.

Obviously, distinct ring extension maps are also distinct as (B, B)-bimodule morphisms, hence F is a faithful functor.

Let be a morphism in YB str_B, where , S∈ Rge_B. Then f is unital, and , hence, for all r, r´∈,

Multiplying factors in tensor products in both sides of the above equation yields 2(f(rr´) − f(r)f(r´)) = 0, hence f(rr´) = f(r)f(r´) and, as f is a (B, B)-bimodule map, it is a ring extension map. Therefore, F is a full functor.

Theorem 3.5 (i) There exists a functor

A coring morphism is mapped into a (B, B)-bimodule morphism.

(ii) G is a full and faithful embbeding.

Proof. i) The proof that ψ_C is a generalised YB operator (cf. Proposition 2.3 from [12]) is left to the reader (ψ_C⁻¹ = ψ_C). Furthermore, for all c∈C, ψ_C(c ⊗ _B 0) = 0 = 0 ⊗ _B c, ψ_C(0 ⊗ _B c) = 0 = c ⊗ _B 0. Moreover, for all c, c´∈C,

and

Hence (C,ψ_C, 0,ε^C) is an object in YB str_B. Let f : C→D be any morphism of B-corings. Then f is also a (B, B)-bimodule morphism, f(0) = 0, ε^D ◦ f = ε^C, and,

Therefore f :(C,ψ_C, 0,ε^C) → (D,ψ_D, 0,ε^D) is a morphism in YB str_B.

(ii) Suppose that GC = GD for some B-corings C, D. This means that C = D as (B, B)-bimodules, ε^C = ε^D, and the only things which can differ are comultiplications. However, as ψ_C = ψ_D, we have

hence

(Δ^C − Δ^D) ⊗ _B ε^C = −ε^C ⊗ _B (Δ^C − Δ^D)

Composing both sides of the above equation with Δ^C yields 2(Δ^C − Δ^D) = 0 hence Δ^C = Δ^D and C = D as (B, B)-corings. Hence G is an embedding.

Obviously distinct B-coring morphisms are also distinct as (B, B)-bimodule morphisms, hence G is a faithful functor.

Let f :(C,ψ_C, 0,ε^C) → (D,ψD, 0,ε^D), where C, D are corings, be a morphism in YB str_B. Then (B, B)-bimodule morphism f : C→D is counital, i.e., ε^D ◦ f = ε^C . Furthermore, (f ⊗ _B f) ◦ ψ_C = ψ_D ◦ (f ⊗ _B f), and hence (f ⊗ _B f) ◦ ψ_C ◦ Δ^C = ψ_D ◦ (f ⊗ _B f) ◦ Δ^C . Observe that ψ_C ◦ Δ^C = Δ^C . Therefore

i.e., 2(f ⊗ _B f) ◦ Δ^C = 2Δ^D ◦ f, hence (f ⊗ _B f) ◦ Δ^C = Δ^D ◦ f, and f is a B-coring map. Therefore G is full.

Proposition 3.6 Let (V, R, e, ε) ∈ r.f.g.pYB str_B. Then

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where e^†(f) = f(e), and

Moreover,

is a natural isomorphism in r.f.g.pYB str_B.

Proof. R is invertible, hence We shall prove that R^† satisfies the Yang–Baxter equation. Observe that

Indeed, let Γ ∈ (V ⊗ _B V )*, f∈ V*, and let be a dual basis of V .

Similarly we can prove the other equality. By virtue of (17,18), we can write

By (2),

and therefore

Hence R^† is a generalised YB operator .

Proofs of bilinearity of e* and centrality of ε are the same as proofs of analogues properties of duals of units and counits in Lemma 2.4. Moreover, for all f∈ V * ,

and

Furthermore, for all x = f ⊗ _B^op g∈ V * ⊗ _B^op V*,

and

Hence (V^*,R^†, ε, e^†) ∈ r.f.g.pYB str_B^op.

Morphism κ : () → ()** is natural in _BB, and as V is finitely generated projective, κ_V is invertible. Therefore it suffices to prove that κ_V is a morphism in r.f.g.pYB str_B. To this end, observe first that

κ_V (e) = f → f(e)^op = e^†

and, for all υ∈ V ,

Note that is a dual basis of V*. Therefore, for all Γ ∈ (V* ⊗ _B^op V*)*,

and so, for all υ, υ´ ∈ V ,

Therefore, κ_V is a morphism in r.f.g.pYB str_B as required.

Proposition 3.7 Let ∈ r.f.g.pRge_B, C∈ r.f.g.pCrg_B. Then i.e.,

Proof. From Lemma 2.4 we know that 1_R^† = ε^R* and 1_C* = ε^C. Furthermore, for all c, c´∈*,

Similarly, for all r, r´∈C* , rr´ = κ_C,C(r ⊗ _B^op r´) ◦ Δ^C, therefore for all r, r´∈C* ,

This completes the proof.

Remark 3.8 Put together the statements of Theorem 3.6, Theorem 3.5, Proposition 3.6 and Proposition 3.7, can be summarized in the following diagram:

This means that the duality between right finitely generated projective ring extensions of B and B corings extends to the category r.f.g.pYB str_B.

We extended the duality between right finitely generated projective ring extensions and right finitely generated projective corings to the category of right finitely generated projective generalized Yang–Baxter structures. This duality and its extension could be seen as a more general construction. For example, at the epistemologic level, the extension of the duality of (co)algebra structures seems to be a model for the relation between interdisciplinarity, pluridisciplinarity and transdisciplinarity (see [4]). It would be interesting to interpret this construction in terms of particle interactions.

The relationships between sub(co)algebras and (co)ideals are well-known, and the term of YB ideal was proposed for the first time in [11]. The following question arises: What are the relationships between sub(co)rings, (co)ideals and generalized Yang–Baxter structures?

We think that there are more connections between the Pontryagin–van Kampen duality and the above extension of the duality of (co)algebra structures.