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The Larson–Sweedler theorem for multiplier Hopf algebras
- Source :
-
Journal of Algebra . Feb2006, Vol. 296 Issue 1, p75-95. 21p. - Publication Year :
- 2006
-
Abstract
- Abstract: 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. [Copyright &y& Elsevier]
- Subjects :
- *MATHEMATICS
*ALGEBRA
*MATHEMATICAL analysis
*ALGEBRAIC topology
Subjects
Details
- Language :
- English
- ISSN :
- 00218693
- Volume :
- 296
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Journal of Algebra
- Publication Type :
- Academic Journal
- Accession number :
- 19358087
- Full Text :
- https://doi.org/10.1016/j.jalgebra.2005.11.020