Polynomial identities and noncommutative versal torsors |
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Authors: | Eli Aljadeff |
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Affiliation: | a Department of Mathematics, Technion - Israel Institute of Technology, 32000 Haifa, Israel b Institut de Recherche Mathématique Avancée, CNRS - Université Louis Pasteur, 7 rue René Descartes, 67084 Strasbourg, France |
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Abstract: | To any cleft Hopf Galois object, i.e., any algebra obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle α, we attach two “universal algebras” and . The algebra is obtained by twisting the multiplication of H with the most general two-cocycle σ formally cohomologous to α. The cocycle σ takes values in the field of rational functions on H. By construction, is a cleft H-Galois extension of a “big” commutative algebra . Any “form” of can be obtained from by a specialization of and vice versa. If the algebra is simple, then is an Azumaya algebra with center . The algebra is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras; it is the universal comodule algebra in which all comodule algebra identities of are satisfied. We construct an embedding of into ; this embedding maps the center of into when the algebra is simple. In this case, under an additional assumption, , thus turning into a central localization of . We completely work out these constructions in the case of the four-dimensional Sweedler algebra. |
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Keywords: | 16R50 16W30 16S35 16S38 16S40 16H05 16E99 17B37 55R10 58B32 81R50 81R60 |
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