Chain and distributive coalgebras

We show that coalgebras whose lattice of right coideals is distributive are coproducts of coalgebras whose lattice of right coideals is a chain. Those chain coalgebras are characterized as finite duals of Noetherian chain rings whose residue field is a finite dimensional division algebra over the base field. They also turn out to be coreflexive. Infinite dimensional chain coalgebras are finite duals of left Noetherian chain domains. Given any finite dimensional division algebra D and D-bimodule structure on D, we construct a chain coalgebra as a cotensor coalgebra. Moreover if D is separable over the base field, every chain coalgebra of type D can be embedded in such a cotensor coalgebra. As a consequence, cotensor coalgebras arising in this way are the only infinite dimensional chain coalgebras over perfect fields. Finite duals of power series rings with coeficients in a finite dimensional division algebra D are further examples of chain coalgebras, which also can be seen as tensor products of D^∗, and the divided power coalgebra and can be realized as the generalized path coalgebra of a loop. If D is central, any chain coalgebra is a subcoalgebra of the finite dual of D[[x]].     

Quasi-co-Frobenius Coalgebras. II

Abstract

We prove new characterizations of Quasi-co-Frobenius (QcF) coalgebras and co-Frobenius coalgebras. Among them, we prove that a coalgebra is QcF if and only if C generates every left and every right C-comodule. We also prove that every QcF coalgebra is Morita-Takeuchi equivalent to a co-Frobenius coalgebra.     

When Does the Rational Torsion Split Off for Finitely Generated Modules

It is well known that the torsion part of any finitely generated module over the formal power series ring K[[X]] is a direct summand. In fact, K[[X]] is an algebra dual to the divided power coalgebra over K and the torsion part of any K[[X]]-module actually identifies with the rational part of that module. More generally, for a certain general enough class of coalgebras—those having only finite dimensional subcomodules—we see that the above phenomenon is preserved: the set of torsion elements of any C ^*-module is exactly the rational submodule. With this starting point in mind, given a coalgebra C we investigate when the rational submodule of any finitely generated left C ^*-module is a direct summand. We prove various properties of coalgebras C having this splitting property. Just like in the K[[X]] case, we see that standard examples of coalgebras with this property are the chain coalgebras which are coalgebras whose lattice of left (or equivalently, right, two-sided) coideals form a chain. We give some representation theoretic characterizations of chain coalgebras, which turn out to make a left-right symmetric concept. In fact, in the main result of this paper we characterize the colocal coalgebras where this splitting property holds non-trivially (i.e. infinite dimensional coalgebras) as being exactly the chain coalgebras. This characterizes the cocommutative coalgebras of this kind. Furthermore, we give characterizations of chain coalgebras in particular cases and construct various and general classes of examples of coalgebras with this splitting property.     

Localization in tame and wild coalgebras

We apply the theory of localization for tame and wild coalgebras in order to prove the following theorem: “Let Q be an acyclic quiver. Then any tame admissible subcoalgebra of KQ is the path coalgebra of a quiver with relations”.     

On radicals of module coalgebras

We introduce the notion of an idempotent radical class of module coalgebras over a bialgebra B. We prove that if R is an idempotent radical class of B-module coalgebras, then every B-module coalgebra contains a unique maximal B-submodule coalgebra in R. Moreover, a B-module coalgebra C is a member of R if, and only if, DB is in R for every simple subcoalgebra D of C. The collection of B-cocleft coalgebras and the collection of H-projective module coalgebras over a Hopf algebra H are idempotent radical classes. As applications, we use these idempotent radical classes to give another proofs for a projectivity theorem and a normal basis theorem of Schneider without assuming a bijective antipode.     

Cofree Compositions of Coalgebras

We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and in the theory of species. We prove that the composition of two cofree coalgebras is again cofree, and we give sufficient conditions that ensure the composition is a one-sided Hopf algebra. We show that these conditions are satisfied when one coalgebra is a graded Hopf operad ${\mathcal{D}}$ and the other is a connected graded coalgebra with coalgebra map to ${\mathcal{D}}$ . We conclude by computing the primitive elements for compositions of coalgebras built on the vertices of multiplihedra, composihedra, and hypercubes.     

Covering Coalgebras and Dual Non-singularity

Localisation is an important technique in ring theory and yields the construction of various rings of quotients. Colocalisation in comodule categories has been investigated by some authors (see Jara et al., Commun. Algebra, 34(8):2843–2856, 2006 and Nastasescu and Torrecillas, J. Algebra, 185:203–220, 1994). Here we look at possible coalgebra covers π : D → C that could play the rôle of a coalgebra colocalisation. Codense covers will dualise dense (or rational) extensions; a maximal codense cover construction for coalgebras with projective covers is proposed. We also look at a dual non-singularity concept for modules which turns out to be the comodule-theoretic property that turns the dual algebra of a coalgebra into a non-singular ring. As a corollary we deduce that hereditary coalgebras and hence path coalgebras are non-singular in the above sense. We also look at coprime coalgebras and Hopf algebras which are non-singular as coalgebras.     

Vertex coalgebras, comodules, cocommutativity and coassociativity

We introduce the notion of vertex coalgebra, a generalization of vertex operator coalgebras. Next we investigate forms of cocommutativity, coassociativity, skew-symmetry, and an endomorphism, D^∗, which hold on vertex coalgebras. The former two properties require grading. We then discuss comodule structure. We conclude by discussing instances where graded vertex coalgebras appear, particularly as related to Primc’s vertex Lie algebra and (universal) enveloping vertex algebras.     

Dual Coalgebras of Jordan Bialgebras and Superalgebras

W. Michaelis showed for Lie bialgebras that the dual coalgebra of a Lie algebra is a Lie bialgebra. In the present article we study an analogous question in the case of Jordan bialgebras. We prove that a simple infinite-dimensional Jordan superalgebra of vector type possesses a nonzero dual coalgebra. Thereby, we demonstrate that the hypothesis formulated by W. Michaelis for Lie coalgebras fails in the case of Jordan supercoalgebras.     

The Kantor-Koecher-Tits construction for Jordan coalgebras

The relationship between Jordan and Lie coalgebras is established. We prove that from any Jordan coalgebra 〈L(A), Δ〉, it is possible to construct a Lie coalgebra 〈L(A), Δ_L〉. Moreover, any dual algebra of the coalgebra 〈L(A), Δ_L〉 corresponds to a Lie algebra that can be determined from the dual algebra for (A, Δ), following the Kantor-Koecher-Tits process. The structure of subcoalgebras and coideals of the coalgebra 〈L(A), Δ_L〉 is characterized. Supported by ISF grant No. RB 6000. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 173–189, March–April, 1996.     

余代数上余倾斜余模的结构

研究了余代数上余倾斜余模的结构特征,证明了每个余倾斜余模都可以写成不可分解的两两非同构的余模的直和形式,每个余倾斜余模包含所有的内射不可分解模作为直和项.最后构造了余倾斜余模的两个例子.     

(H,R)-Lie coalgebras and (H,R)-Lie bialgebras

Given an (H,R)-Lie coalgebra Γ, we construct (H,R _T )-Lie coalgebra Γ^T through a right cocycle T, where (H,R) is a triangular Hopf algebra, and prove that there exists a bijection between the set of (H,R)-Lie coalgebras and the set of ordinary Lie coalgebras. We also show that if (L, [, ], Δ, R) is an (H,R)-Lie bialgebra of an ordinary Lie algebra then (L ^T , [, ], Δ_T, R _T ) is an (H,R _T )-Lie bialgebra of an ordinary Lie algebra.     

Generators for comonoids and universal constructions

We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or Hopf algebra. We find concrete generators for the categories of coalgebras in these monoidal categories, and explicitly construct cofree coalgebras, products and limits of coalgebras in each case. This answers an open question in Agore (Proc Am Math Soc 139:855–863, 2011) on the existence of a cofree coring, and constructs the cofree (co)module coalgebra on a B-(co)module, for a bialgebra B.     

广义路余代数和广义对偶Gabriel定理

通过将箭图的每个顶点放置一个k-余代数,首先引进了广义路余代数的概念,其次给出了广义路余代数的一些基本性质,还讨论了同构问题.证明了两个正规广义路余代数是同构的当且仅当他们的箭图及对应顶点上的单余代数是同构的.对于满足Codim C_0■1余代数C,证明了对偶Wedderburn-Malcev定理成立.作为广义路余代数的一个应用,推广了点余代数的对偶Gabriel定理.     

Maps into homotopy coalgebras

Let X be a k-fold homotopy coalgebra of order j with respect to the pair of adjoint functors Σk and Ωk. We show that, under some connectivity conditions on the map , Y inherits a k-fold homotopy coalgebra structure of the same order for which f is a morphism of homotopy coalgebras. In particular, this holds for skeleta of homotopy coalgebras under some mild assumptions. As a consequence, we complete results on [M. Arkowitz, M. Golasiński, Homotopy coalgebras and k-fold suspensions, Hiroshima Math. J. 27 (1997) 209-220] and [T. Ganea, Cogroups and suspensions, Invent. Math. 9 (1970) 185-197] by detecting k-fold suspensions among skeleta of k-fold homotopy coalgebras.     

Serial coalgebras and their valued Gabriel quivers

We study serial coalgebras by means of their valued Gabriel quivers. In particular, Hom-computable and representation-directed serial coalgebras are characterized. The Auslander–Reiten quiver of a serial coalgebra is described. Finally, a version of Eisenbud–Griffith Theorem is proved, namely, every subcoalgebra of a prime, hereditary and strictly quasi-finite coalgebra is serial.     

(*)-Serial coalgebras

In this paper, we introduce the notion of (*)-serial coalgebras which is a generalization of serial coalgebras. We investigate the properties of (*)-serial coalgebras and their comodules, and obtain sufficient and necessary conditions for a basic coalgebra to be (*)-serial.     

The strong Brauer group of a cocommutative coalgebra

Using strong equivalences for coalgebras we define the strong Brauer group of a cocommutative coalgebra C, which is a subgroup of the Brauer group of C. In general there is not a good relation between the Brauer group of a coalgebra and the Brauer group of the dual algebra C∗, the former is not even a torsion group. We find that this subgroups embeds in the Brauer group of C∗. A key tool in this result is the use of techniques from torsion theory. Some cases where both subgroups coincide are shown, for example, C being coreflexive.     

On extensions of rational modules

Given a topological algebra A, we investigate when the categories of all rational A-modules and of finite-dimensional rational modules are closed under extensions inside the category of A-modules. We give a complete characterization of these two properties, in terms of a topological and a homological condition, for complete algebras. We also give connections to other important notions in coalgebra theory such as coreflexive coalgebras. In particular, we are able to generalize many previously known partial results and answer some questions in this direction, and obtain large classes of coalgebras for which rational modules are closed under extensions as well as various examples where this is not true.