The Structure of Frobenius Algebras and Separable Algebras |
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Authors: | S. Caenepeel Bogdan Ion G. Militaru |
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Affiliation: | (1) Faculty of Applied Sciences, Free University of Brussels, VUB, Pleinlaan 2, B-1050 Brussels, Belgium;(2) Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08544-1000, U.S.A.;(3) Faculty of Mathematics, University of Bucharest, Str. Academiei 14, RO-70109 Bucharest 1, Romania |
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Abstract: | We present a unified approach to the study of separable and Frobenius algebras. The crucial observation is that both types of algebras are related to the nonlinear equation R12R23=R23R13=R13R12, called the FS-equation. Solutions of the FS-equation automatically satisfy the braid equation, an equation that is in a sense equivalent to the quantum Yang–Baxter equation. Given a solution to the FS-equation satisfying a certain normalizing condition, we can construct a Frobenius algebra or a separable algebra A(R) – the normalizing condition is different in both cases. The main result of this paper is the structure of these two fundamental types of algebras: a finite dimensional Frobenius or separable k-algebra A is isomorphic to such an A(R). A(R) can be described using generators and relations. A new characterization of Frobenius extensions is given: B A is Frobenius if and only if A has a B-coring structure (A, , ) such that the comultiplication : A AB A is an A-bimodule map. |
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Keywords: | separable algebra Frobenius algebra quantum Yang– Baxter equation |
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