Separability and the Twisted Frobenius Bimodule |
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Authors: | Lars Kadison |
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Affiliation: | (1) Chalmers Tekniska Högskola och Göteborgs Universitet, S-412 96 Göteborg, Sweden |
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Abstract: | This paper begins with an introduction to -Frobenius structure on a finite-dimensional Hopf subalgebra pair. In Section 2 a study is made of a generalization of Frobenius bimodules and -Frobenius extensions. Also a special type of twisted Frobenius bimodule which gives an endomorphism ring theorem and converse is studied. Section 3 brings together material on separable bimodules, the dual definitions of split and separable extension, and a theorem of Sugano on endomorphism rings of separable bimodules. In Section 4, separable twisted Frobenius bimodules are characterized in terms of data that generalizes a Frobenius homomorphism and a dual base. In the style of duality, two corollaries characterizing split -Frobenius and separable -Frobenius extensions are proven. Sugano"s theorem is extended to -Frobenius extensions and their endomorphism rings. In Section 5, the problem of when separable extensions are Frobenius extensions is discussed. A Hopf algebra example and a matrix example are given of finite rank free separable -Frobenius extensions which are not Frobenius in the ordinary sense. |
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Keywords: | Frobenius bimodules /content/l867jll9gwm78431/xxlarge946.gif" alt=" beta" align=" MIDDLE" BORDER=" 0" >-Frobenius extensions separable extensions split extensions Hopf algebras endomorphism ring theorem |
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