Coalgebraic coalgebras
HTML articles powered by AMS MathViewer
- by D. E. Radford PDF
- Proc. Amer. Math. Soc. 45 (1974), 11-18 Request permission
Abstract:
We investigate coalgebras $C$ over a field $k$ such that the dual algebra ${C^ \ast }$ is an algebraic algebra ($C$ is called coalgebraic). The study reduces to the cosemisimple and connected cases. If $C$ is co-semisimple and coalgebraic, then ${C^ \ast }$ is of bounded degree. If $C$ is connected, then $C$ is coalgebraic if, and only if, every coideal is the intersection of cofinite coideals. Our main result is that if $C$ is a coalgebra over an infinite field $k$ and the Jacobson radical $\operatorname {Rad} {C^ \ast }$ is nil, there is an $n$ such that ${a^n} = 0$ all $a \in \operatorname {Rad} {C^ \ast }$. By the Nagata-Higman theorem, $\operatorname {Rad} {C^ \ast }$ is nilpotent if nil in characteristic 0.References
-
I. N. Herstein, Theory of rings, Math. Lecture Note Series, University of Chicago, Chicago, Ill., 1961.
- Robert G. Heyneman and David E. Radford, Reflexivity and coalgebras of finite type, J. Algebra 28 (1974), 215–246. MR 346001, DOI 10.1016/0021-8693(74)90035-0
- Robert G. Heyneman and Moss Eisenberg Sweedler, Affine Hopf algebras. I, J. Algebra 13 (1969), 192–241. MR 245570, DOI 10.1016/0021-8693(69)90071-4
- David E. Radford, Coreflexive coalgebras, J. Algebra 26 (1973), 512–535. MR 327818, DOI 10.1016/0021-8693(73)90012-4
- Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 45 (1974), 11-18
- MSC: Primary 16A24
- DOI: https://doi.org/10.1090/S0002-9939-1974-0357474-9
- MathSciNet review: 0357474