Galois Theory for H-extensions and H-coextensions

Details Galois Theory for H-extensions and H-coextensions Galois Theory for H-extensions and H-coextensions

2009-12-09

arxiv.org/abs/0912.1795v7

Mathematics

Quantum Algebra

Left admissibility for subalgebras changed, an assumption added to Theorem 4.7 (A^{op} is H^{op}-Galois) and proof of Theorem

Scientific paper

We show that there exists a Galois correspondence between subalgebras of an H-comodule algebra A over a base ring R and generalised quotients of a Hopf algebra H. We also show that Q-Galois subextensions are closed elements of the constructed Galois connection. Then we consider the theory of coextensions of H-module coalgebras. We construct Galois theory for them and we prove that H-Galois coextensions are closed. We apply the obtained results to the Hopf algebra itself and we show a simple proof that there is a bijection correspondence between right ideal coideals of H and its left coideal subalgebras when H is finite dimensional. Furthermore we formulate necessary and sufficient conditions when the Galois correspondence is a bijection for arbitrary Hopf algebras. We also present new conditions for closedness of subalgebras and generalised quotients when A is a crossed product.

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