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Generalized Frobenius Algebras and Hopf Algebras

Published online by Cambridge University Press:  20 November 2018

Miodrag Cristian Iovanov
Miodrag Cristian Iovanov*
Affiliation:
University of Southern California, Department of Mathematics, 3620 South Vermont Ave. KAP 108, Los Angeles, California 90089-2532/ e-mail: yovanov@gmail.com iovanov@usc.edu
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Abstract

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“Co-Frobenius” coalgebras were introduced as dualizations of Frobenius algebras. We previously showed that they admit left-right symmetric characterizations analogous to those of Frobenius algebras. We consider the more general quasi-co-Frobenius $\left( \text{QcF} \right)$ coalgebras. The first main result in this paper is that these also admit symmetric characterizations: a coalgebra is $\text{QcF}$ if it is weakly isomorphic to its (left, or right) rational dual $\text{Rat}\left( {{C}^{*}} \right)$ in the sense that certain coproduct or product powers of these objects are isomorphic. Fundamental results of Hopf algebras, such as the equivalent characterizations of Hopf algebras with nonzero integrals as left (or right) co-Frobenius, $\text{QcF}$, semiperfect or with nonzero rational dual, as well as the uniqueness of integrals and a short proof of the bijectivity of the antipode for such Hopf algebras all follow as a consequence of these results. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras. Furthermore, we introduce a general concept of Frobenius algebra, which makes sense for infinite dimensional and for topological algebras, and specializes to the classical notion in the finite case. This will be a topological algebra $A$ that is isomorphic to its complete topological dual ${{A}^{\vee }}$. We show that $A$ is a (quasi)Frobenius algebra if and only if $A$ is the dual ${{C}^{*}}$ of a (quasi)co-Frobenius coalgebra $C$. We give many examples of co-Frobenius coalgebras and Hopf algebras connected to category theory, homological algebra and the newer $q$-homological algebra, topology or graph theory, showing the importance of the concept.

Keywords

16T1518G3516T0520N9918D1005E10coalgebraHopf algebraintegralFrobeniusQcFco-Frobenius.
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 1 , February 2014 , pp. 205 - 240
Copyright
Copyright © Canadian Mathematical Society 2014

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