Co-Frobenius coalgebras

Iovanov, Miodrag Cristian

doi:10.1016/j.jalgebra.2006.04.028

Cited by 21 publications

(26 citation statements)

References 18 publications

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“…This adds to the previously known symmetric characterization of co-Frobenius coalgebras from [11], where it is shown that C is co-Frobenius if and only if C is isomorphic to its left (or, equivalently, to its right) rational dual Rat( C * C * ). Moreover, it is shown there that this is further equivalent to the functors C * -dual Hom C * (−, C * ) and K-dual Hom K (−, K) from C * M to K M being isomorphic when restricted to the category Rat( C * M) of left (equivalently, right) rational C * -modules which is the same as the category M C of right C-comodules (Rat( C * M) = M C ).…”

Section: Abstract Algebraic Integrals and Frobenius Categoriesmentioning

confidence: 62%

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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Section: Abstract Algebraic Integrals and Frobenius Categoriesmentioning

confidence: 62%

Abstract Algebraic Integrals and Frobenius Categories

Iovanov

2013

Int. J. Math.

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“…(ii)⇒(i) is easy, since any finitely generated left submodule of M * is closed (this is a well known fact; one can also see [I,Lemma 1…”

Section: Noetherian and Artinian Objectsmentioning

confidence: 85%

Triangular matrix coalgebras and applications

Iovanov

2013

Linear and Multilinear Algebra

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Abstract. We formally introduce and study generalized comatrix coalgebras and upper triangular comatrix coalgebras, which are not only a dualization but also an extension of classical generalized matrix algebras. We use these to answer several open questions on Noetherian and Artinian type notions in the theory of coalgebras, and to give complete connections between these. We also solve completely the so called finite splitting problem for coalgebras: we show that C is a coalgebra such that the rational part of every left finitely generated C * -module splits off if and only if C = D M 0 E is an upper triangular matrix coalgebra, for a serial coalgebra D whose Ext-quiver is a finite union of cycles, a finite dimensional coalgebra E and a finite dimensional bicomodule M .

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“…We note that this also follows directly: : if C * is semilocal, then every simple module is rational, since C * /J is finite dimensional semisimple. By [I,Lemma 1.4], if A = C * is a profinite algebra dual to the coalgebra C, every simple rational left C * -module S has a projective cover; it is obtained via the canonical morphism E(S * ) * → S. This shows that C * is semiperfect by an equivalent. A basic question would be to understand simple modules over such an algebra A = C * .…”

Section: Semiartinian Profinite Algebrasmentioning

confidence: 99%