Twisted hopf comodule algebras

For k a commutative ring, H a k‐bialgebra and A a right H‐comodule k‐algebra, we define a new multiplication on the H‐comodule A to obtain a twisted algebra” A^T, T sumHom(H,End (A)). If ^T is convolution invertible, the categories of relative right Hopf modules over A and A^Tare isomorphic. Similarly a convolution invertible left twisting gives an isomorphism of the categories of relative left Hopf modules. We show that crossed products are invertible twistings of the tensor product, and obtain, as a corollary, a duality theorem for crossed products     

On identities for coalgebras and hopf algebras

The notion of a polynomial identity for algebras is well-known. In this work the notion of identity is transferred to coalgebras (where one may call it a coidentity). The author studies the basic properties of coidenti-ties, some examples of Hopf algebras with an identity or a coidentity are considered.     

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.     

Centrally closed hopf module algebras

We introduce a concept of centrally closed Hopf module algebras and show that centrally closed Hopf module algebras inherit all basic properties of centrally closed algebras.     

On the Projectivity of module coalgebras

In this paper, we derive some criteria for the projectivity of a module coalgebra over a finite dimensional Hopf algebra. In particular, we show that any Hopf algebra over a field of characteristic zero is faithfully flat over its group-like subHopf algebra. Finally, we prove that if is a finite dimensional subHopf algebra of a Hopf algebra , then is normal in if and only if . This improves a result by S. Montgomery (1993).

     

The structure theorem for weak module coalgebras

Let H be a weak Hopf algebra, let C be a weak right H-module coalgebra, and let $ \bar C = {C \mathord{\left/ {\vphantom {C C}} \right. \kern-0em} C} \cdot Ker \sqcap ^L $ . We prove a structure theorem for weak module coalgebras, namely, C is isomorphic as a weak right H-module coalgebra to a weak smash coproduct $ \bar C $ × H defined on a k-space $$ \{ \Sigma c_{(0)} \otimes h_2 \varepsilon (c_{( - 1)} h_1 )|c \in C,h \in H\} $$ if there exists a weak right H-module coalgebra map ?: C → H.     

弱模余代数的结构定理

设H为弱Hopf代数,C为弱右H-模余代数,令C=C/C·ker L.利用Smash余积来研究弱模余代数上的结构定理,并给出了C与C×H作为余代数同构的条件.     

Evolution coalgebras

We introduce the notion of evolution coalgebra as an algebraic structure to describe the backwards evolution of non-Mendelian genetic systems and establish their connection with evolution algebras. We study their main properties and consider evolution coalgebras endowed with genetic realization.     

Structurable coalgebras

The notion of a structurable coalgebra is defined. It is proved that every structurable coalgebra is locally finite-dimensional. Supported by RFFR grant No. 95-01-01356 and by ISF grant No. RB 6000. Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 529–542, September–October, 1996.     

(*)-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.     

On Gorenstein coalgebras

We investigate the comodule representation category over the Morita-Takeuchi context coalgebra Γ and study the Gorensteinness of Γ. Moreover, we determine explicitly all Gorenstein injective comodules over the Morita-Takeuchi context coalgebra Γ and discuss the localization in Gorenstein coalgebras. In particular, we describe its Gabriel quiver and carry out some examples when the Morita-Takeuchi context coalgebra is basic.     

Relative homological coalgebras

We study classes of relative injective and projective comodules and extend well-known results about projective comodules given in [7]. The existence of covers and envelopes by these classes of comodules is also studied and used to characterize the projective dimension of a coalgebra. We also compare this homological coalgebra with the very intensively studied homological algebra of the dual algebra (see [5]). This revised version was published online in June 2006 with corrections to the Cover Date.     

Self-dual hopf algebras

We introduce the notions of self-dual (graded) Hopf algebras and of structurally simple (graded) Hopf algebras. We prove that the self-dual Hopf algebras are structurally simple and provide a construction of self-dual Hopf algebras. Finally, we classify the self-dual quotients of the form T_B (M)/I, where T_B (M) is a path algebra with a graded Hopf algebra structure, and I is a graded admissible Hopf ideal.     

Products of coalgebras

We prove that the category of F-coalgebras is complete, that is products and equalizers exist, provided that the type functor F is bounded or preserves mono sources. This generalizes and simplifies a result of Worrell ([Wor98]). We also describe the relationship between the product and the largest bisimulation between and and find an example of two finite coalgebras whose product is infinite. Received January 11, 2000; accepted in final form October 16, 2000.     

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.