Chain and distributive coalgebras

• Type: Article in International Scientific Journal
• Date: 2007

Authors

lomp, christian (Author) FCUP; sant'ana, alveri (Author) Other

Journal

Title: Journal of Pure and Applied Algebra

Vol. 211 No. 3

Pages: 581-595

ISSN: 0022-4049

Publisher: Elsevier

Indexing

ISI Web of Knowledge - 6 Citations

Scopus - 6 Citations

Other information

Authenticus ID: P-004-63B

DOI: 10.1016/j.jpaa.2007.02.009

Abstract (EN): We show that coalgoebras whose lattice of right coideals is distributive are coproducts of coalgebras whose lattice of right coideals is a chain. Those chain coalgebras are characterized as finite duals of Noetherian chain rings whose residue field is a finite dimensional division algebra over the base field. They also turn out to be coreflexive. Infinite dimensional chain coalgebras are finite duals of left Noetherian chain domains. Given any finite dimensional division algebra D and D-bimodule structure on D, we construct a chain coalgebra as a cotensor coalgebra. Moreover if D is separable over the base field, every chain coalgebra of type D can be embedded in such a cotensor coalgebra. As a consequence, cotensor coalgebras arising in this way are the only infinite dimensional chain coalgebras over perfect fields. Finite duals of power series rings with coeficients in a finite dimensional division algebra D are further examples of chain coalgebras, which also can be seen as tensor products of D*, and the divided power coalgebra and can be realized as the generalized path coalgebra of a loop. If D is central, any chain coalgebra is a subcoalgebra of the finite dual of D[[x]].

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 15