Semisimple Hopf actions on commutative domains

Etingof, Pavel; Walton, Chelsea

doi:10.1016/j.aim.2013.10.008

Cited by 65 publications

(84 citation statements)

References 11 publications

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“…This work contributes to the field of noncommutative invariant theory in the sense of studying quantum analogues of group actions on commutative k-algebras. Here, we restrict our attention to the actions of finite quantum groups, i.e., finite dimensional Hopf algebras, as these objects and their actions on (quantum) k-algebras have been the subject of recent research in noncommutative invariant theory, including [8], [10], [16], [18], [27], [29], [34], [35], [37]. The two important classes of finite dimensional Hopf algebras H are those that are semisimple (as a k-algebra) and those that are pointed (namely, all simple H-comodules are 1-dimensional).…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, we have many choices of what one could consider to be a quantum k-algebra, but from the viewpoint of classical invariant theory and algebraic geometry, the examination of Hopf actions on commutative domains over k is of interest. Since the classification of semisimple Hopf actions on commutative domains over k is understood by the work of the authors [18], the focus of this article is to classify finite dimensional non-semisimple Hopf (H-) actions on commutative domains over k, particularly when H is pointed.…”

Section: Introductionmentioning

confidence: 99%

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Pointed Hopf Actions on Fields, I

Etingof

Walton

2015

Transformation Groups

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Actions of semisimple Hopf algebras H over an algebraically closed field of characteristic zero on commutative domains were classified recently by the authors in [18]. The answer turns out to be very simple -if the action is inner faithful, then H has to be a group algebra. The present article contributes to the non-semisimple case, which is much more complicated. Namely, we study actions of finite dimensional (not necessarily semisimple) Hopf algebras on commutative domains, particularly when H is pointed of finite Cartan type.The work begins by reducing to the case where H acts inner faithfully on a field; such a Hopf algebra is referred to as Galois-theoretical. We present examples of such Hopf algebras, which include the Taft algebras, uq(sl 2 ), and some Drinfeld twists of other small quantum groups. We also give many examples of finite dimensional Hopf algebras which are not Galois-theoretical. Classification results on finite dimensional pointed Galois-theoretical Hopf algebras of finite Cartan type will be provided in the sequel, Part II, of this study.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Pointed Hopf Actions on Fields, I

Etingof

Walton

2015

Transformation Groups

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“…By Proposition 2(3) one concludes that H has to be cocommutative, and hence a group algebra. Their key result of [3] extending [4] is the following: As a consequence from this Proposition and the reduction process to fields of positive characteristic, as described in sections 2 and 3, one deduces: …”

Section: Reduction To Hopf Algebras Over Finite Fieldsmentioning

confidence: 94%

A note on a paper by Cuadra, Etingof and Walton

Lomp

Pansera

2016

Communications in Algebra

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Abstract. We analyse the proof of the main result of a paper by Cuadra, Etingof and Walton, which says that any action of a semisimple Hopf algebra H on the nth Weyl algebra A = An(K) over a field K of characteristic 0 factors through a group algebra. We verify that their methods can be used to show that any action of a semisimple Hopf algebra H on an iterated Ore extension of derivation typeThe purpose of this note is to analyse the main result of the paper [3] which says that any action of a semisimple Hopf algebra H on the nth Weyl algebra A = A n (K) over a field K of characteristic 0 factors through a group algebra. The central idea is to pass from algebras in characteristic 0 to algebras in positive characteristic by using the subring R of K, generated by all structure constants of H and the action on A and by passing to a finite field R/m.It has already been outlined in [3, p.2] that these methods could be used to establish more general results on semisimple Hopf actions on quantized algebras and that the authors of [3] will do so in their future work. In particular it has been announced in [3, p.2] that their methods will apply to actions on module algebras A such that the resulting algebra A p when passing to a field of characteristic p, for large p, is PI and their PI-degree is a power of p. Such algebras include universal enveloping algebras of finite dimensional Lie algebras and algebras of differential operators of smooth irreducible affine varieties.We will verify part of their outlined program in Theorem 6 and will show that any action of a semisimple Hopf algebra H on an enveloping algebra of a finite dimensional Lie algebra or on an iterated Ore extension of derivation typein characteristic zero factors through a group algebra. Apart from this we give an alternative proof for the reduction step in Proposition 2.

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“…Note that when the action of H preserves the standard filtration of A n (k), Theorem 4.1 can be deduced from [EW1,Proposition 5.4], since the associated graded algebra gr(A n (k)) is a commutative domain. Our main achievement in this paper is to eliminate this assumption.…”

Section: Introductionmentioning

confidence: 99%

“…Let k be an algebraically closed field of characteristic zero and H a semisimple Hopf algebra over k. In [EW1,Theorem 1.3], two of the authors showed that any action of H on a commutative domain over k factors through a group action. The goal of this paper is to extend this result to Weyl algebras.…”

Section: Introductionmentioning

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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Semisimple Hopf actions on commutative domains

Cited by 65 publications

References 11 publications

Pointed Hopf Actions on Fields, I

Pointed Hopf Actions on Fields, I

A note on a paper by Cuadra, Etingof and Walton

Semisimple Hopf actions on Weyl algebras

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