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We study the duality between corings and ring extensions. We construct a new category with a self-dual functor acting on it, which extends that duality. This construction can be seen as the non-commutative case of another duality extension: the duality between finite dimensional algebras and coalgebra. Both these duality extensions have some similarities with the Pontryagin-van Kampen duality theorem.

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 [

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 [

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

In

In

There are some more comments to be made.

(i) We propose as a research project the investigation of other connections between the duality of (co)algebras and the Pontryagin duality. (For example, one might try to endow the (co)algebra structures with some topological structures.)

(ii) 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 [

(iii) This paper explains that taking the dual of some objects can be seen a “continuous” process. Let us visualize this statement by considering an example from geometry. We take a triangular prism: We can see it as two parallel triangles joint by 3 segments. In total it has 5 planar geometric figures, 9 edges and 6 vertices. The geometric dual of the triangular prism has 6 planar geometric figures, 9 edges and 5 vertices. Now, one can start with a triangular prism, “shave” its corners, and then continuously deform that figure in order to obtain the geometric dual of the triangular prism.

Throughout this paper 𝕂 is a commutative ring, and all 𝕂-modules

Let _{A}_{B}_{A}_{B}

For all _{A}_{B}

We denote by (·)^{op }: ^{op }the canonical anti-algebra isomorphism from the algebra ^{op }(which is the identity on the underlying 𝕂-modules), ^{op }as module elements and (^{op} = ^{op}^{op }for all

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

(i) If _{A}_{B}_{A}^{op}_{B}^{op}^{op}^{op})(

Assume that _{A}_{B}

(ii) The mapping _{M}_{M}^{op }is an isomorphism in _{A}_{B}_{A}_{B}^{** }: _{A}_{B}_{A}_{B}

(iii) If _{B}_{C}_{M,N}_{B}^{op}_{B} N_{M,N}_{B}^{op}_{A}^{op}_{C}^{op}

(iv) Let _{A}_{B}_{B}_{C}_{C}_{D}

(v) Let _{A}_{B}_{B}_{C}_{B} N_{A}_{C}

The following terminology and theorems concerning corings and ring extensions are needed in this paper. For a review on coalgebras see: [

_{B}_{B}^{C}^{C}_{B}_{B}^{C}_{B} C^{C}

In the sequel we shall use Sweedler’s notation Δ^{C}_{(1)} ⊗ _{B} c_{(2)}. Given _{B}_{B}_{B}^{C }^{D}^{D}^{C}_{B}

_{B}_{B}_{B}_{B}_{B}

The full subcategory of _{B}_{B}_{B}_{B}

_{B}_{B}^{op}_{B}^{op}^{op }with multiplication

_{C}_{*} := ^{C}

_{B}^{op}-coring with comultiplication and counit

^{op}-corings.

_{B}_{B}_{B}. For all _{B}, _{R}

_{B}_{B}_{B}. For all _{B}, κ_{C}

(

and

Hence

(_{C}_{B}

Hence, for all

and

_{B}_{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: [

Let _{B}_{B}^{12 }= _{B}^{23 }= _{B}_{B}_{B}_{B}_{B}

_{B}_{B}

_{B}

(i)

(ii) _{B}_{B}

(iii) _{B}_{B}_{B}

(iv) _{B}_{B}_{B}_{B}

A morphism _{B}

(v) (_{B}_{B}

(vi)

(vii) ^{´ }◦

Composition of morphisms is defined as the standard composition of _{B}_{B}

_{B}_{B}_{B}

_{R}_{R}_{R}_{B}

Let _{R}_{S}

Hence _{B}

(_{B}_{S}_{R}_{R}_{S}

hence

Multiplying tensor factors on both sides of this equation (whether using multiplication in

Obviously, distinct ring extension maps are also distinct as (

Let _{B}_{B}

Multiplying factors in tensor products in both sides of the above equation yields 2(

_{C}_{C}^{−1 }= _{C}_{C}_{B}_{B}_{C}_{B}_{B}

and

Hence (_{C}^{C}_{B}^{D}^{C}

Therefore _{C}^{C}_{D}^{D}_{B}

(^{C }^{D}_{C}_{D}

hence

(Δ^{C}^{D}_{B}^{C}^{C}_{B}^{C}^{D}

Composing both sides of the above equation with Δ^{C}^{C}^{D}^{C}^{D}

Obviously distinct

Let _{C}^{C}^{D}_{B}^{D}^{C}_{B}_{C}_{D}_{B}_{B}_{C}^{C}_{D}_{B}^{C}_{C}^{C}^{C}

_{B}^{C}^{D}_{B}^{C}^{D}

_{B}. Then

^{†}(f) = f(e),

_{B}.

^{† }satisfies the Yang–Baxter equation. Observe that

Indeed, let Γ ∈ (_{B}

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

By (2),

and therefore

Hence ^{† }is a generalised YB operator .

Proofs of bilinearity of

and

Furthermore, for all _{B}^{op}_{B}^{op}

and

Hence (^{*},^{†}, ^{†}) ∈ _{B}^{op}

Morphism _{B}_{B}_{V}_{V}_{B}

_{V}^{op }= ^{† }

and, for all

Note that _{B}^{op}

and so, for all

Therefore, _{V}_{B}

_{B}, C_{B}. Then

_{R}^{†} = ^{R*}_{C*}^{C}

Similarly, for all _{C,C}_{B}^{op}^{C}

This completes the proof.

This means that the duality between right finitely generated projective ring extensions of B and B corings extends to the category _{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 [

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 [

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

We would like to thank Tomasz Brzezi´nski for helpful remarks. The first author thanks for a Marie Curie Research Fellowship, HPMF-CT-2002-01782 at Swansea University. The work of BZ was supported by the EPSRC grant GR/S01078/01.