## Abstract

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

Original language | English |
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Pages (from-to) | 1453-1495 |

Number of pages | 43 |

Journal | Advances in Mathematics |

Volume | 218 |

Issue number | 5 |

DOIs | |

State | Published - 1 Aug 2008 |

Externally published | Yes |

## Keywords

- Cocycle
- Comodule algebra
- Galois extension
- Hopf algebra
- Polynomial identity