Abstract
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.
Highlights
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
(i) We propose as a research project the investigation of other connections between the duality ofalgebras and the Pontryagin duality. (For example, one might try to endow thealgebra structures with some topological structures.)
(ii) At the epistemologic level, the extension of the duality ofalgebra structures seems to be a model for the relation between interdisciplinarity, pluridisciplinarity and transdisciplinarity
Summary
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. 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]. We show that taking the right dual is a duality functor in the category of right finitely generated projective generalized Yang–Baxter structures. 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. (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,. One can start with a triangular prism, “shave” its corners, and continuously deform that figure in order to obtain the geometric dual of the triangular prism
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