2004/08/17 by Alfons Van Daele, Van Daele, Alfons, Shuanhong Wang +1
Mathematics · #FOS: Mathematics #Quantum Algebra (math.QA) #math.QA
paper · pdf · doi:10.48550/arxiv.math/0408218
arxiv created 2004/08/17 · arxiv updated 2009/12/01
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.