The double algebraic view of finite quantum groupoids

Abstract

Double algebra is the structure modelled by the properties of the ordinary and the convolution product in Hopf algebras, weak Hopf algebras and Hopf algebroids if a Frobenius integral is given. The Hopf algebroids possessing a Frobenius integral are precisely the Frobenius double algebras in which the two multiplications satisfy distributivity. The double algebra approach makes it manifest that all comultiplications in such measured Hopf algebroids are of the Abrams-Kadison type, i.e., they come from a Frobenius algebra structure in some bimodule category. Antipodes for double algebras correspond to the Connes-Moscovici 'deformed' antipode as we show by discussing Hopf and weak Hopf algebras from the double algebraic point of view. Frobenius algebra extensions provide further examples that need not be distributive.

Integral Theory for Hopf Algebroids

“…A Maschke-type theorem on certain Hopf algebroids can be obtained also by the application of [37,Theorem 4.2]. Notice, however, that the Hopf algebroids occurring this way are only the Frobenius Hopf algebroids (discussed in Section 4 below), that is the Hopf algebroids possessing nondegenerate integrals (which are called Frobenius integrals in [37] The following assertions on a Hopf algebroid A = (A L , A R , S) are equivalent: [17,Proposition 2.6] that a separable extension is both left and right semi-simple.…”

“…The Hopf algebroids, satisfying the equivalent conditions of Theorem 4.7, provide examples of distributive Frobenius double algebras [37]. (Notice that the integrals, which we call nondegenerate, are called Frobenius integrals in [37].…”

Finitary Galois extensions over noncommutative bases

“…Therefore if η is mono (e.g., if M R is faithful) then we conclude from [27,Theorem 4.2] that the k-algebra H is a separable extension of B or, equivalently, of T . Note that in the presence of the Frobenius condition left D2 is equivalent to right D2 and in the presence of the D2 Frobenius condition M N is balanced iff N M is balanced.…”

“…It turns out [27,Proposition 3.2] that vertical multiplication with the horizontal type of comultiplications ∆ B and ∆ T obey bialgebroid like relations. However, in Frobenius DAs the comultiplications need not be multiplicative.…”

“…By the results of [27,Section 8.11] this is equivalent to the assumption that V is a Frobenius Hopf algebroid and H op is its dual.…”