The Goldie Theorem for H-semiprime algebras

Skryabin, Serge; Oystaeyen, Freddy Van

doi:10.1016/j.jalgebra.2006.06.030

Cited by 35 publications

(49 citation statements)

References 42 publications

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“…Proof The H-semisimplicity of A is proved in [24,Lemma 4.2] (this follows easily also from Theorem 1.1 whose proof does not depend on Theorem 2.8). Thus A is isomorphic to A 1 × · · · × A n where each A i is an H-simple H-module algebra.…”

Section: Theorem 28 Let a Be An Artinian H-semiprime H-module Algebrmentioning

confidence: 90%

Hopf Algebra Orbits on the Prime Spectrum of a Module Algebra

Skryabin¹

2008

Algebr Represent Theor

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For a Hopf algebra H and an H-module algebra A module-finite over its center it is proved that there is an equivalence relation on a subset Spec f A of the prime spectrum of A which exactly corresponds to the orbit relation in case of group actions. A linearly compact topologically H-simple H-module algebra L P (A) have been associated with each P ∈ Spec f A. When A is noetherian and H-semiprime, it is shown that A has a quasi-Frobenius classical quotient ring.

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Section: Theorem 28 Let a Be An Artinian H-semiprime H-module Algebrmentioning

confidence: 90%

Hopf Algebra Orbits on the Prime Spectrum of a Module Algebra

Skryabin¹

2008

Algebr Represent Theor

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“…Then U ∼ = (U ′ ) * for some U ′ ∈ M H , dim U ′ < ∞. By Lemma 4.6 M ∼ = U ′ ⊗ V , so M is finitely generated in M A according to [29,Lemma 1.1]. The direct summand M n of M is also finitely generated in M A .…”

Section: Proof (I) This Is the Isomorphism From [29 Lemma 11] It mentioning

confidence: 95%

Projectivity of Hopf algebras over subalgebras with semilocal central localizations

Skryabin¹

2008

J K-Theor

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The purpose of this paper is to extend the class of pairs A, H where H is a Hopf algebra over a field and A a right coideal subalgebra for which H is proved to be either projective or flat as an A-module. The projectivity is obtained under the assumption that H is residually finite dimensional, A has semilocal localizations with respect to a central subring, and there exists a Hopf subalgebra B of H such that the antipode of B is bijective and B is a finitely generated A-module. The flatness of H over A is shown to hold when H is a directed union of residually finite dimensional Hopf subalgebras, and there exists a Hopf subalgebra of H whose center contains A. More general projectivity and flatness results are established for (co)equivariant modules over an H-(co)module algebra under similar assumptions.

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“…Since B is a PI domain, after localization we obtain a classical quotient ring Q B , which is a division algebra; see [C,Corollary 7.5.2] and [GW,Theorem 6.8]. Now apply [SV,Theorem 2.2] to obtain the result. (ii) If H is semisimple and cosemisimple, any action of H on D factors through an action of a cocommutative Hopf algebra.…”

Section: Semisimple Cosemisimple Hopf Actions On Division Algebrasmentioning

confidence: 99%

“…Also, B ⊂ Q ρ,Ap , and by (1), h r · B ⊂ B for all r. Thus, the map ρ ′ defines a coaction B → B ⊗ H * p . By [SV,Theorem 2.2] (which applies because B is contained in a division algebra), ρ ′ extends to a coaction…”

Section: Proposition 44 Proposition 43 Remains True If the Module mentioning

confidence: 99%

Semisimple Hopf actions on Weyl algebras

Cuadra

Etingof

Walton

2015

Advances in Mathematics

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Abstract. We study actions of semisimple Hopf algebras H on Weyl algebras A over an algebraically closed field of characteristic zero. We show that the action of H on A must factor through a group action; in other words, if H acts inner faithfully on A, then H is cocommutative. The techniques used include reduction modulo a prime number and the study of semisimple cosemisimple Hopf actions on division algebras.

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The Goldie Theorem for H-semiprime algebras

Cited by 35 publications

References 42 publications

Hopf Algebra Orbits on the Prime Spectrum of a Module Algebra

Hopf Algebra Orbits on the Prime Spectrum of a Module Algebra

Projectivity of Hopf algebras over subalgebras with semilocal central localizations

Semisimple Hopf actions on Weyl algebras

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