Polynomial identities and noncommutative versal torsors

TY - JOUR

T1 - Polynomial identities and noncommutative versal torsors

AU - Aljadeff, Eli

AU - Kassel, Christian

PY - 2008/8/1

Y1 - 2008/8/1

N2 - To any cleft Hopf Galois object, i.e., any algebra αH obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle α, we attach two "universal algebras" AHα and UHα. The algebra AHα 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, AHα is a cleft H-Galois extension of a "big" commutative algebra BHα. Any "form" of αH can be obtained from AHα by a specialization of BHα and vice versa. If the algebra αH is simple, then AHα is an Azumaya algebra with center BHα. The algebra UHα 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 αH are satisfied. We construct an embedding of UHα into AHα; this embedding maps the center ZHα of UHα into BHα when the algebra αH is simple. In this case, under an additional assumption, AHα ≅ BHα ⊗ZHα UHα, thus turning AHα into a central localization of UHα. We completely work out these constructions in the case of the four-dimensional Sweedler algebra.

AB - To any cleft Hopf Galois object, i.e., any algebra αH obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle α, we attach two "universal algebras" AHα and UHα. The algebra AHα 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, AHα is a cleft H-Galois extension of a "big" commutative algebra BHα. Any "form" of αH can be obtained from AHα by a specialization of BHα and vice versa. If the algebra αH is simple, then AHα is an Azumaya algebra with center BHα. The algebra UHα 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 αH are satisfied. We construct an embedding of UHα into AHα; this embedding maps the center ZHα of UHα into BHα when the algebra αH is simple. In this case, under an additional assumption, AHα ≅ BHα ⊗ZHα UHα, thus turning AHα into a central localization of UHα. We completely work out these constructions in the case of the four-dimensional Sweedler algebra.

KW - Cocycle

KW - Comodule algebra

KW - Galois extension

KW - Hopf algebra

KW - Polynomial identity

UR - http://www.scopus.com/inward/record.url?scp=44249093261&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2008.03.014

DO - 10.1016/j.aim.2008.03.014

M3 - 文章

AN - SCOPUS:44249093261

SN - 0001-8708

VL - 218

SP - 1453

EP - 1495

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 5

ER -