Integrals in Hopf Algebras over Rings

Lomp, Christian

doi:10.1081/agb-200036837

Cited by 12 publications

(12 citation statements)

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“…Since the counit of a Hopf algebra H over a commutative ring k is a split epimorphism of k-modules, the Maschke theorem generalizes to this case in the following form [12,20]. The extension k → H is separable if and only if it is (left and right) semi-simple and if and only if there exist normalized (left and right) integrals in H .…”

Section: Maschke Type Theoremsmentioning

confidence: 93%

“…This is equivalent to the true semi-simplicity of H (i.e. the true projectivity of any H -module [28]) if and only if k is a semi-simple ring [20].…”

Section: Introductionmentioning

confidence: 97%

“…The integrals in Hopf algebras over principal ideal domains were analyzed in [19,32] where the following -by now classical -results have been proven: There are numerous generalizations of these results in the literature. Historically the first is due to Pareigis [27] who proved the following statements on a finitely generated and projective Hopf algebra The generalization of the Maschke theorem to Hopf algebras H over commutative rings k states that the existence of a normalized left integral in H is equivalent to the separability of H over k, which is further equivalent to its relative semisimplicity in the sense of [15,17] that any H -module is (H, k)-projective [12,20]. This is equivalent to the true semi-simplicity of H (i.e.…”

Section: Introductionmentioning

confidence: 99%

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Integral Theory for Hopf Algebroids

Böhm

2005

Algebr Represent Theor

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The theory of integrals is used to analyze the structure of Hopf algebroids. We prove that the total algebra of a Hopf algebroid is a separable extension of the base algebra if and only if it is a semi-simple extension and if and only if the Hopf algebroid possesses a normalized integral. It is a Frobenius extension if and only if the Hopf algebroid possesses a nondegenerate integral. We give also a sufficient and necessary condition in terms of integrals, under which it is a quasi-Frobenius extension, and illustrate by an example that this condition does not hold true in general. Our results are generalizations of classical results on Hopf algebras. (2000): 16W30. Mathematics Subject Classification

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Section: Maschke Type Theoremsmentioning

confidence: 93%

“…This is equivalent to the true semi-simplicity of H (i.e. the true projectivity of any H -module [28]) if and only if k is a semi-simple ring [20].…”

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Integral Theory for Hopf Algebroids

Böhm

2005

Algebr Represent Theor

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“…we have a ring homomorphism H R → End R (A R ). Since t ∈ H R still satisfies ǫ(t) = 1, H R is separable over R. Analogously as t * ∈ H * R satisfies t * (1) = 1, H * R is separable over R (see [7,9]). Note that H ≃ H R ⊗ R K and A ≃ A R ⊗ R K as algebras.…”

Section: Inner Faithful Actionmentioning

confidence: 99%

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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“…The map ϕ M : A left integral of an Hopf algebra is an element t ∈ H such that ht = (h)t for all h ∈ H. Left integrals are related to the finiteness of the Hopf algebra (see [14]). A module algebra A is said to have an element of trace 1 if there exists an element a ∈ A and a left integral t such that t · a = 1.…”

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confidence: 99%

Prime Elements in Partially Ordered Groupoids Applied to Modules and Hopf Algebra Actions

Lomp

2005

J. Algebra Appl.

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Primeness on modules can be defined by prime elements in a suitable partially ordered groupoid. Using a product on the lattice of submodules L(M ) of a module M defined in [3] we revise the concept of prime modules in this sense. Those modules M for which L(M ) has no nilpotent elements have been studied by Jirasko and they coincide with Zelmanowitz' "weakly compressible" modules. In particular we are interested in representing weakly compressible modules as a subdirect product of "prime" modules in a suitable sense. It turns out that any weakly compressible module is a subdirect product of prime modules (in the sense of Kaplansky). Moreover if M is a self-projective module, then M is weakly compressible if and only if it is a subdirect product of prime modules (in the sense of Bican et al.). An application to Hopf actions is given.

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Integrals in Hopf Algebras over Rings

Cited by 12 publications

References 12 publications

Integral Theory for Hopf Algebroids

Integral Theory for Hopf Algebroids

A note on a paper by Cuadra, Etingof and Walton

Prime Elements in Partially Ordered Groupoids Applied to Modules and Hopf Algebra Actions

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