Integral Theory for Hopf Algebroids

The theory of integrals is used to analyze the structure of Hopf algebroids. We prove that the total algebra of a Hopf algebroid is a separable extension of the base algebra if and only if it is a semi-simple extension and if and only if the Hopf algebroid possesses a normalized integral. It is a Frobenius extension if and only if the Hopf algebroid possesses a nondegenerate integral. We give also a sufficient and necessary condition in terms of integrals, under which it is a quasi-Frobenius extension, and illustrate by an example that this condition does not hold true in general. Our results are generalizations of classical results on Hopf algebras.     

A theorem on the endomorphism ring of reduced torsion-free abelian groups and some applications

Sunto Si espone una condiztone sufficiente affinchè un anello topologico sia l'anello degli endomorfismi, dotato della topologia finita, di un gruppo abeliano ridotto, senza torsione e di cardinalità ≤. Il risultato ottenuto estende un teorema di A. L. S. Corner ([3] Theorem1.1) ed un teorema di A. Orsatti ([9] Teorema A*).

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Entrata in Redazione il 17 febbraio 1977.

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Lavoro eseguito nell'ambito dei Gruppi di Ricerca Matematica del C.N.R.     

Very weak solutions to elliptic equations with singular convection term

We study Dirichlet problem for a nonlinear equation with a drift term. Despite the presence of the singular convection term, we establish existence and uniqueness of a solution in spaces larger than the natural one.     

The weak theory of monads

We construct a ‘weak’ version EMw(K) of Lack and Street's 2-category of monads in a 2-category K, by replacing their compatibility constraint of 1-cells with the units of monads by an additional condition on the 2-cells. A relation between monads in EMw(K) and composite pre-monads in K is discussed. If K admits Eilenberg-Moore constructions for monads, we define two symmetrical notions of ‘weak liftings’ for monads in K. If moreover idempotent 2-cells in K split, we describe both kinds of weak lifting via an appropriate pseudo-functor EMw(K)→K. Weak entwining structures and partial entwining structures are shown to realize weak liftings of a comonad for a monad in these respective senses. Weak bialgebras are characterized as algebras and coalgebras, such that the corresponding monads weakly lift for the corresponding comonads and also the comonads weakly lift for the monads.