Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras

Abstract

We extend the Larson-Sweedler theorem [Amer. J. Math. 91 (1969) 75] to weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a non-degenerate left integral. We show that the category of modules over a weak Hopf algebra is autonomous monoidal with semisimple unit and invertible modules. We also reveal the connection of invertible modules to left and right grouplike elements in the dual weak Hopf algebra. Defining distinguished left and right grouplike elements, we derive the Radford formula [Amer. J. Math. 98 (1976) 333] for the fourth power of the antipode in a weak Hopf algebra and prove that the order of the antipode is finite up to an inner automorphism by a grouplike element in the trivial subalgebra A T of the underlying weak Hopf algebra A.

Integral Theory for Hopf Algebroids

“…Similar results are known also for the generalizations of Hopf algebras. Integrals for finite-dimensional quasi-Hopf algebras [14] over fields were studied in [16,25,26,11] and for finite-dimensional weak Hopf algebras [4,3] over fields in [3,40].…”

The double algebraic view of finite quantum groupoids

“…Therefore (see p. 510 of [21]) (1) ) λ, S −1 a (1) a (2) = a (3) π L S −1 a (2) a (1) λ, S −1 a (1) a (2) = S S −1 a (3) S −1 a (2) a (1) λ, S −1 a (1) a (2) = a (1) λ, S −1 a a (2) . (8.14)…”

“…The dual basis of ϕ T does not follow easily from this because S does not relate them like it did ϕ L and ϕ R . It is time to introduce S := ν −1 S −1 by analogy with the Hopf case, where ν is the Nakayama automorphism of λ, therefore [21] ν = S −2 α where α(a) := σ a. So we have S(a) = σ −1 S(a), an algebra antiautomorphism of W .…”

On the Structure of Weak Hopf Algebras

“…This is a straightforward dualization of (22). Equation (23) The group G 0 ðHÞ of all trivial group-like elements is a normal subgroup in GðHÞ: Define * G GðHÞ ¼ GðHÞ=G 0 ðHÞ; the quotient group of GðHÞ by G 0 ðHÞ: Let g/ * g g denote the canonical projection from GðHÞ to * G GðHÞ: It turns out that * G GðHÞ plays a more important role than GðHÞ; as it possesses properties extending those of the group of group-like elements of a usual Hopf algebra.…”