The structure theorem and duality theorem for endomorphism algebras of weak Hopf algebras

With an aim of exploring homological algebra for weak Hopf modules, this paper investigates the HOM-functor and presents the structure theorem for endomorphism algebras of weak two-sided (A,H)-Hopf modules, and gives the duality theorem for weak “big” smash products.     

Stable endomorphism algebras of modules over special biserial algebras

 We prove that the stable endomorphism algebra of a module without self-extensions over a special biserial algebra is a gentle algebra. In particular, it is again special biserial. As a consequence, any algebra which is derived equivalent to a gentle algebra is gentle. Received: 17 May 2002 / Published online: 16 May 2003 Mathematics Subject Classification (2000): 16E30, 16G20, 18E30. We gratefully acknowledge support from the Volkswagen Foundation (RIP Program at Oberwolfach). The first author thanks the Nuffield Foundation (Grant Number NAL/00270/G) for financial support.     

Extensions of the endomorphism algebra of weak comodule algebras

Let H be a weak Hopf algebra, and let A/B be a weak right H-Galois extension. In this paper, we mainly discuss the extension of the endomorphism algebra of a module over A. A necessary and sufficient condition for such an extension of the endomorphism algebra to be weak H-Galois is obtained by using Hopf-Galois theory and Morita theory.     

Doi-hopf modules over weak hopf algebras

The theorv of Doi-Hopf modules [8,11] is generalized to Weak Hopf Algebras [1, 14, 2].     

A maschke type theorem for weak Hopf algebras

In this paper, we give a necessary and sufficient condition for a comodule algebra over a weak Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the Hopf algebra setting. Also, from these results, we deduce a version of Maschke＇s Theorem for （H, B）-Hopf modules associated with a weak Hopf algebra H and a right H-comodule algebra B.     

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.     

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.     

The structure theorem for comodule algebras over Hopf algebroids

We obtain the structure theorem for $(H, \mathcal{H})$ -Hopf bimodules over Hopf algebroids, where H is the total algebra of the Hopf algebroid  $\mathcal{H}$ . Based on this theorem, we investigate the structure theorem for comodule algebras over Hopf algebroids.     

The diagram for endomorphism algebras of two-term tilting complexes over self-injective algebras

We construct a diagram between the endomorphism algebra of a two-term projective module complex over a self-injective algebra and the endomorphism algebras of its homology groups and apply the diagram to the tilting theory of self-injective algebras. We give an example of recovering the endomorphism algebra of a two-term tilting complex over a self-injective algebra from the endomorphism algebras of its homology groups with some additional structures which appear in the diagram.     

A structure theorem for quasi-Hopf comodule algebras

If is a quasi-Hopf algebra and is a right -comodule algebra such that there exists a morphism of right -comodule algebras, we prove that there exists a left -module algebra such that . The main difference when comparing to the Hopf case is that, from the multiplication of , which is associative, we have to obtain the multiplication of , which in general is not; for this we use a canonical projection arising from the fact that becomes a quasi-Hopf -bimodule.

     

A Maschke type theorem for weak group entwined modules and applications

Let π be a discrete group. Given a weak π-entwining structure \({(A,C)_{\pi - \psi }}\) and α ∈ π, we give the necessary and sufficient conditions for the forgetful functor \({F^{(\alpha )}}\) from the category \(U_A^{\pi - C}(\psi )\) of right \({(A,C)_{\pi - \psi }}\) -modules to the category \({M_{{A_\alpha }}}\) of right \({A_\alpha }\) -modules to be separable. This leads to a generalized notion of integrals. The results are applied to weak Doi-Hopf π-modules and to weak entwining modules.     

Fundamental theorem of projective geometry in Lie modules and algebras

Collineations and lattice isomorphisms for modules over principal ideal rings (generally noncommutative) are studied, and also analogs of the fundamental theorem of protective geometry are proved for a number of classes of Lie algebras.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 18, pp. 165–187, 1986.     

Maschke-type theorem and Morita context over weak Hopf algebras

This paper gives a Maschke-type theorem over semisimple weak Hopf algebras, extends the well-known Maschke-type theorem given by Cohen and Fishman and constructs a Morita context over weak Hopf algebras.