A ug 2 00 4 The Larson - Sweedler theorem for multiplier Hopf algebras
نویسندگان
چکیده
Any finite-dimensional Hopf algebra has a left and a right integral. Conversely, Larsen and Sweedler showed that, if a finite-dimensional algebra with identity and a comultiplication with counit has a faithful left integral, it has to be a Hopf algebra. In this paper, we generalize this result to possibly infinite-dimensional algebras, with or without identity. We have to leave the setting of Hopf algebras and work with multiplier Hopf algebras. Moreover, whereas in the finite-dimensional case, there is a complete symmetry between the bialgebra and its dual, this is no longer the case in infinite dimensions. Therefore we consider a direct version (with integrals) and a dual version (with cointegrals) of the Larson-Sweedler theorem. We also add some results about the antipode. Furthermore, in the process of this paper, we obtain a new approach to multiplier Hopf algebras with integrals.
منابع مشابه
Larson–Sweedler Theorem, Grouplike Elements and Invertible Modules in Weak Hopf Algebras
We extend the Larson–Sweedler theorem for 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 establish the autonomous monoidal category of the modules of a weak Hopf algebra A and show the semisimplicity of the unit and the invertible modules of A. We also reveal the connection of these modules to lef...
متن کاملLarson–Sweedler Theorem, Grouplike Elements, Invertible Modules and the Order of the Antipode in Weak Hopf Algebras
We extend the Larson–Sweedler theorem for 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 establish the autonomous monoidal category of the modules of a weak Hopf algebra A and show the semisimplicity of the unit and the invertible modules of A. We also reveal the connection of these modules to lef...
متن کاملLarson–Sweedler Theorem and the Role of Grouplike Elements in Weak Hopf Algebras
We extend the Larson–Sweedler theorem [10] 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 grouplik...
متن کاملA ug 2 00 6 A note on Radford ’ s S 4 formula
In this note, we show that Radford's formula for the fourth power of the antipode can be proven for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This of course not only includes the case of a finite-dimensional Hopf algebra but also the case of any Hopf algebra with integrals (co-Frobenius Hopf algebras). The proof follows in a few lines from well-known formula...
متن کاملA ug 2 00 5 Multiplier Hopf group coalgebras from algebraic and analytical point of views
The Multiplier Hopf Group Coalgebra was introduced by Hegazi in 2002 [] as a generalization of Hope group caolgebra, introduced by Turaev in 2000 [], in the non-unital case. We prove that the concepts introduced by A.Van Daele in constructing multiplier Hopf algebra [3] can be adapted to serve again in our construction. A multiplier Hopf group coalgebra is a family of algebras A = {A α } α∈π , ...
متن کامل