An Associative Orthogonal Bilinear Form for Hopf Algebras

Abstract

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Burnside’s theorem for Hopf algebras

“…Furthermore, since A/1 is semisimple, it follows that y must be the set of all simple ^4/7-modules. D Our final consequence uses the fact that any finite-dimensional Hopf algebra A is a Fróbenius algebra [LS,§5] and hence that every simple ^4-module is isomorphic to a minimal left ideal of A .…”

Frobenius Functors of the Second Kind

“…We also show that Frobenius ring extensions of the second kind ( [16], [25]) provide examples of Frobenius pairs of the second kind. We use this machinery in Section 3 to give a generalization of the Larson-Sweedler result that a finite dimensional Hopf algebra over a field k is a Frobenius extension of k ( [21]): we show that a finitely generated projective Hopf algebra over a commutative ring is always Frobenius of the second type; as a consequence, we obtain that the integral spaces of the Hopf algebra and its dual are isomorphic. The results of Section 2 can be dualized to functors between categories of comodules; following Takeuchi's approach [30], we give a coalgebra version of Morita's result, and use this to characterize Frobenius pairs of the second kind between categories of comodules.…”

Remarks on Quantum Integration

“…Integrals on (finite dimensional) Hopf algebras have been studied extensively, see for example [12,19] and references therein. For the braided case see the treatment in [13,14] for the basics of the theory -some examples appear in [3,10].…”