168 research outputs found
Galois Theory for H-extensions and H-coextensions
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.Comment: Left admissibility for subalgebras changed, an assumption added to
Theorem 4.7 (A^{op} is H^{op}-Galois) and proof of Theorem 4.7 adde
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