On extensions of rational modules

Iovanov, Miodrag Cristian

doi:10.1007/s11856-013-0070-3

Cited by 2 publications

(18 citation statements)

References 29 publications

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“…(ii) It remains to prove the surjectivity of φ l . Since C 0 is coreflexive and C is right semiperfect, by [9,Corollary 4.10] we see that C is coreflexive. Let C = j∈J E(S j ) be the decomposition into indecomposable left comodules as before.…”

Section: Applications To Coalgebrasmentioning

confidence: 78%

“…Let H be the algebra of functions on G. Proof. If L is a finite dimensional Lie algebra over C, by [6], U (L) is coreflexive, since the coradical of U (L) is 1-dimensional and the space of primitives is finite dimensional (see also the results of [9,Sections 2 & 4]). With the notations for G and H as above, since H 0 = C[G]⊗U (L) as coalgebras.…”

Section: 1mentioning

confidence: 99%

“…One motivation is the duality between commutative Hopf algebras and algebraic groups, as noted above: G −→ H = O(G) = algebra of functions on G, H −→ G = G(H 0 ). In fact the above mentioned example of coreflexive Hopf algebra in [18] holds more generally for every finite dimensional Lie algebra L. Namely, using results in [6,9], it is not difficult to see that if L is a finite dimensional (say over C), U (L) is coreflexive. Moreover, with notations for G and H as above, we see that H 0 = G U (L) is a coreflexive coalgebra too (see Proposition 4.7).…”

Section: Introductionmentioning

confidence: 99%

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Semiperfect and coreflexive coalgebras

Dăscălescu

Iovanov

2013

Forum Mathematicum

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Abstract. We study non-counital coalgebras and their dual non-unital algebras, and introduce the finite dual of a non-unital algebra. We show that a theory that parallels in good part the duality in the unital case can be constructed. Using this, we introduce a new notion of left coreflexivity for counital coalgebras, namely, a coalgebra is left coreflexive if C is isomorphic canonically to the finite dual of its left rational dual Rat(C * C * ). We show that right semiperfectness for coalgebras is in fact essentially equivalent to this left reflexivity condition, and we give the connection to usual coreflexivity. As application, we give a generalization of some recent results connecting dual objects such as quiver or incidence algebras and coalgebras, and show that Hopf algebras with non-zero integrals (compact quantum groups) are coreflexive. Introduction and PreliminariesLet K be a field and let Γ be a quiver. Two combinatorial objects that are important in representation theory are associated with Γ: the quiver algebra K[Γ], and the path coalgebra KΓ. The quiver algebra is an algebra with enough idempotents, but does not have a unit unless Γ is finite, while the path coalgebra KΓ is a coalgebra with counit. In [5] it was investigated how one of the two objects can be recovered from the other one; a finite dual coalgebra A 0 was constructed for an algebra A with enough idempotents, and it was proved that KΓ is isomorphic to K[Γ] 0 provided that Γ has no oriented cycles and between any two vertices of Γ there are finitely many arrows. On the other hand, the algebra K[Γ] can be recovered as the left (or right) rational part of (KΓ) * , provided that for any vertex v of Γ there are finitely many paths starting at v and finitely many paths ending at v (see [5]). A consequence of these results is that KΓ is coalgebra isomorphic to ((KΓ) * rat ) 0 for certain general enough Γ. Parallel results are obtained for incidence (co)algebras of a locally finite partially ordered set X, in which case the role of KΓ, (KΓ) * and K[Γ] is played by the incidence coalgebra KX, the incidence algebra IA(X), and the finite incidence algebra F IA(X), consisting of functions in IA(X) of finite support. It is natural to ask whether a general context that unifies these structures and results exists, for arbitrary algebras and coalgebras. The first step in obtaining such a context is having a well behaved duality for non-unital algebras and coalgebras, that would naturally include the (co)unital case. We extend the construction of the finite dual coalgebra A 0 to the case of an arbitrary algebra A, not necessarily with enough idempotents, and extend the coalgebra theory to coalgebras without counit. We show that several results on coalgebras, including the fundamental theorem of coalgebras, carry over to non-counital coalgebras. These are needed and used for the above mentioned unification, but may also present some interest in their own. Some of the results from coalgebra theory can be extended to the non-counital case with parallel pro...

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Section: Applications To Coalgebrasmentioning

confidence: 78%

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Semiperfect and coreflexive coalgebras

Dăscălescu

Iovanov

2013

Forum Mathematicum

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“…One problem of a particular interest is deciding when is this subcategory of rational modules closed under extensions. This has been considered before by many authors [4,6,11,14,10,19,20,26,28,33]. In general, given an abelian or Grothendieck category A and a closed subcategory B of A, one can consider the trace functor T with respect to B, which is defined as T (M ) =the sum of all subobjects of M which belong to B.…”

Section: Introductionmentioning

confidence: 99%

“…However, the problem of finding an equivalent characterization for when rational modules are closed under extensions remained open. Also mainly motivated by the interesting developments in [6], such an equivalent characterization was obtained in [14,Theorem 3.7], in the form of a topological condition on the open ideals of C * and a homological Ext condition. Using this result in part, a generalization to arbitrary coalgebras of the above equivalent characterizations for the situation when C 0 is finite dimensional is given in [14,Theorem 4.8].…”

Section: Introductionmentioning

confidence: 99%

Complete Path Algebras and Rational Modules

Iovanov

2015

Preprint

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On extensions of rational modules

Cited by 2 publications

References 29 publications

Semiperfect and coreflexive coalgebras

Semiperfect and coreflexive coalgebras

Complete Path Algebras and Rational Modules

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