Multiplier Hopf and bi-algebras

Details Multiplier Hopf and bi-algebras Multiplier Hopf and bi-algebras

2009-01-20

arxiv.org/abs/0901.3103v2

Mathematics

Rings and Algebras

23 pages

Scientific paper

We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele in [A. Van Daele, Multiplier Hopf algebras, {\em Trans. Amer. Math. Soc.}, {\bf 342}(2), (1994) 917--932.] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly non-unital, idempotent, non-degenerate, $k$-projective) algebra over a commutative ring $k$ is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of $k$-modules, into a diagram of strict monoidal forgetful functors.

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