Co-Frobenius Hopf Algebras: Integrals, Doi–Koppinen Modules and Injective Objects

Dăscălescu, S.; Năstăsescu, C.; Torrecillas, Blas

doi:10.1006/jabr.1999.7934

Cited by 13 publications

(11 citation statements)

References 19 publications

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“…The above corollary was also proved in [6], under the supplementary assumption that C is also right semiperfect. In view of our later Theorem 3.4, this condition can, in fact, be dropped.…”

Section: Proposition 15 Let C Be a Left Qcf Coalgebra And σ : T → Smentioning

confidence: 65%

“…In fact, as noted in [14], the parallel between Hopf algebra and compact groups is closer, and the bijectivity of the antipode of a Hopf algebra with nonzero integral can be proved independent of the uniqueness of the integral, which then follows in the same way as the original proof of the uniqueness of the Haar measure on a compact group. Also, the complete reducibility of representations of compact groups can be deduced algebraically from a wellknown result of Sullivan on Hopf algebras in [21] (see also [6] for an alternate proof). …”

Section: C Iovanovmentioning

confidence: 99%

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Abstract Algebraic Integrals and Frobenius Categories

Iovanov

2013

Int. J. Math.

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We generalize results on the connection between existence and uniqueness of integrals and representation theoretic properties for Hopf algebras and compact groups. For this, given a coalgebra C, we study analogues of the existence and uniqueness properties for the integral functor Hom C (C, −), which generalizes the notion of integral in a Hopf algebra. We show that the coalgebra C is co-Frobenius if and only if dim(Hom C (C, M )) = dim(M ) for all finite dimensional right (left) comodules M . As applications, we give a few new categorical characterizations of co-Frobenius, quasi-co-Frobenius (QcF) coalgebras and semiperfect coalgebras, and re-derive classical results of Lin, Larson, Sweedler and Sullivan on Hopf algebras. We show that a coalgebra is QcF if and only if the category of left (right) comodules is Frobenius, generalizing results from finite dimensional algebras, and we show that a one-sided QcF coalgebra is twosided semiperfect. We also construct a class of examples derived from quiver coalgebras to show that the results of the paper are the best possible. Finally, we examine the case of compact groups, and note that algebraic integrals can be interpreted as certain skew-invariant measure theoretic integrals on the group. Abstract Algebraic Integrals and Frobenius Categoriesuniqueness of integrals on the one hand, and representation theoretic properties of C (categorical properties of the category of comodules) on the other, should be possible for general coalgebras.For a coalgebra C and a finite dimensional right C-comodule M we consider the functor l,M = Hom C (C, M ) = Hom C * (C C , M C ) -the space of left integrals, and similarly, for left C-comodules N let r,N = Hom C (C, N ) = Hom C * (C C , C N ) be the space of right integrals (right C * -module morphisms). As noted, this definition has been considered before in literature by several authors [8,7,22,4]. Certain extensions of the above mentioned results for Hopf algebras were, in fact, noticed before. We recall that a coalgebra C is called left (right) co-Frobenius if C embeds in C * as left (right) C * -modules, and simply co-Frobenius if it is both left and right co-Frobenius. A first connection between properties of the integral functor and co-Frobenius coalgebras is proved in [4, Chap. 5.4]: if C is a left and right co-Frobenius coalgebra, then dim(Hom C * (C, M )) ≤ dim(M ) for every left (or right) C-comodule M . This result was proved in [22] for certain classes of co-Frobenius coalgebras (finite dimensional, or cosemisimple, or for co-Frobenius coalgebras which are Hopf algebras).It is natural to think to the dimensional comparison dim( l,M ) ≤ dim(M ) as a "uniqueness" of integrals for M and then to the statement dim( l,M ) ≥ dim(M ) as "existence of integrals". We first show in Sec. 1, that for a coalgebra which is (just) left co-Frobenius, the uniqueness of (left) integrals holds for all right Ccomodules M (dim( l,M ) ≤ dim(M )) and the existence of (right) integrals holds as well for all left C-comodules N (dim( r,N ) ≥ dim(N ))....

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“…The above corollary was also proved in [6], under the supplementary assumption that C is also right semiperfect. In view of our later Theorem 3.4, this condition can, in fact, be dropped.…”

Section: Proposition 15 Let C Be a Left Qcf Coalgebra And σ : T → Smentioning

confidence: 65%

Section: C Iovanovmentioning

confidence: 99%

Abstract Algebraic Integrals and Frobenius Categories

Iovanov

2013

Int. J. Math.

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“…It is easily verified that µ M and ν M are injective linear maps. Inspired by [9,11] we propose the following…”

Section: Doi-hopf Modules and Comodules Over A Coringmentioning

confidence: 99%

“…In Section 2 we look at the case where C is infinite dimensional. Following the methods developed in [9,11,19], we will present two characterizations of the category of Doi-Hopf modules C M(H) B . We first introduce the notion of rational (right) C * ◮<B-module, and then we will show that the category C M(H) B is isomorphic to Rat(M C * ◮<B ), the category of rational (right) C * ◮<B-modules.…”

Section: Introductionmentioning

confidence: 99%

Doi–Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

Bulacu

Caenepeel

Torrecillas

2006

Communications in Algebra

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For a quasi-Hopf algebra H, a left H-comodule algebra B and a right H-module coalgebra C we will characterize the category of Doi-Hopf modules C M(H) B in terms of modules. We will also show that for an Hbicomodule algebra A and an H-bimodule coalgebra C the category of generalized Yetter-Drinfeld modules A YD(H) C is isomorphic to a certain category of Doi-Hopf modules. Using this isomorphism we will transport the properties from the category of Doi-Hopf modules to the category of generalized Yetter-Drinfeld modules.1991 Mathematics Subject Classification. 16W30. He would like to thank Free University of Brussels (Belgium) and University of Almeria (Spain) for their warm hospitality.Using the first isomorphism, we will characterize A YD(H) C as a category of comodules over a coring. In Section 3.3, we will characterize the category of Yetter-Drinfeld modules as a category of modules.A 2 M(H op ⊗ H) C . It will follow from Proposition 3.4 that these two categories are isomorphic. Proposition 3.4. Let H be a quasi-bialgebra, C a left H-module coalgebra andProof. If A 1 and A 2 are twist equivalent, then there exists V ∈ A ⊗ H satisfying (1.22-1.24). Take M ∈ A 1 M(H) C ; M becomes an object in A 2 M(H) C by keeping the same left A-module structure and defining ρ ′ M : M → M ⊗ C, ρ ′ (m) = V · ρ M (m). Conversely, take M ∈ A 2 M(H) C via the structures · and ρ ′ M . Then M can be viewed as a left-right (H, A 1 , C)-Hopf module via the same left A-action · and the right C-coaction ρ M defined by ρ M : M → M ⊗ C, ρ M (m) = V −1 · ρ ′ M (m).

show abstract

“…Note that in all cases mentioned in the result above, C is left and right co-Frobenius. In [7,Theorem 4.4] it is proved that for any right co-Frobenius coalgebra C which is also left semiperfect, we have that dim l,C ,M dim M for any finite dimensional left C -comodule M. In [15, Corollary 1.8] Iovanov succeeded to prove this result in the case where C is only right co-Frobenius (thus to drop the left semiperfect condition), and moreover to prove that for such a C we also have that dim N dim r,C ,N for any finite dimensional right C -comodule N.…”

mentioning

confidence: 99%

On the dimension of the space of integrals on coalgebras

2010

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We study the injective envelopes of the simple right C -comodules, and their duals, where C is a coalgebra. This is used to give a short proof and to extend a result of Iovanov on the dimension of the space of integrals on coalgebras. We show that if C is right coFrobenius, then the dimension of the space of left M-integrals on C is dim M for any left C -comodule M of finite support, and the dimension of the space of right N-integrals on C is dim N for any right C -comodule N of finite support. Some examples of integrals are computed for incidence coalgebras.

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Co-Frobenius Hopf Algebras: Integrals, Doi–Koppinen Modules and Injective Objects

Cited by 13 publications

References 19 publications

Abstract Algebraic Integrals and Frobenius Categories

Abstract Algebraic Integrals and Frobenius Categories

Doi–Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

On the dimension of the space of integrals on coalgebras

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