Polynomial identities and noncommutative versal torsors

• Autores: Eli Aljadeff, Christian Kassel
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 218, Nº 5, 2008, págs. 1453-1495
• Idioma: inglés
• DOI: 10.1016/j.aim.2008.03.014
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• Resumen

  • To any cleft Hopf Galois object, i.e., any algebra obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle a, we attach two �universal algebras� and . The algebra is obtained by twisting the multiplication of H with the most general two-cocycle s formally cohomologous to a. The cocycle s 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