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Generalized Frobenius Algebras and Hopf Algebras


Published:20130206
Printed: Feb 2014
Miodrag Cristian Iovanov,
University of Southern California, Department of Mathematics, 3620 South Vermont Ave. KAP 108, Los Angeles, California 900892532
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Abstract
"CoFrobenius" coalgebras were introduced as dualizations of
Frobenius algebras.
We previously showed
that they admit
leftright symmetric characterizations analogue to those of Frobenius
algebras. We consider the more general quasicoFrobenius (QcF)
coalgebras; the first main result in this paper is that these also
admit symmetric characterizations: a coalgebra is QcF if it is weakly
isomorphic to its (left, or right) rational dual $Rat(C^*)$, 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) coFrobenius, 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)coFrobenius coalgebra $C$. We
give many examples of coFrobenius coalgebras and Hopf algebras
connected to category theory, homological algebra and the newer
qhomological algebra, topology or graph theory, showing the
importance of the concept.
Keywords: 
coalgebra, Hopf algebra, integral, Frobenius, QcF, coFrobenius
coalgebra, Hopf algebra, integral, Frobenius, QcF, coFrobenius

MSC Classifications: 
16T15, 18G35, 16T05, 20N99, 18D10, 05E10 show english descriptions
Coalgebras and comodules; corings Chain complexes [See also 18E30, 55U15] Hopf algebras and their applications [See also 16S40, 57T05] None of the above, but in this section Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] Combinatorial aspects of representation theory [See also 20C30]
16T15  Coalgebras and comodules; corings 18G35  Chain complexes [See also 18E30, 55U15] 16T05  Hopf algebras and their applications [See also 16S40, 57T05] 20N99  None of the above, but in this section 18D10  Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23] 05E10  Combinatorial aspects of representation theory [See also 20C30]
