Weak Corings

✍ Scribed by Robert Wisbauer

• Book ID: 102574791
• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 230 KB
• Volume: 245
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2001.8923

✦ Synopsis

Entwined structures A, C, were introduced by Brzezinski and Majid to study the interdependence of an R-algebra A and an R-coalgebra C, R a commutative ring. It turned out that this relationship can also be expressed by the fact that A m C has a canonical A-coring structure. More generally, weak entwined R structures and their modules were studied by Caenepeel and de Groot and Caenepeel suggested relating these to pre-corings. Slightly modifying this notion we introduce weak corings and develop a general theory of comodules over such Ž . corings. In particular we obtain that A, C, is a weak entwined structure if and Ž . only if A m C is a weak A-coring with canonical structure maps . Weak bialge-R bras in the sense of Bohm᎐Nill᎐Szlachanyi are characterized as R-modules with an ¨Ž . algebra and coalgebra structure B, , ⌬ such that B m B is a weak coring for R the various coring structures induced by , ( , ⌬, and ( ⌬. Moreover we will characterize weak Hopf algebras as those weak bialgebras B, which are generators Ž . for the comodules over B m B и 1.