The Larson-Sweedler theorem and the operator algebra approach to quantum groups
The Larson-Sweedler theorem says that a bialgebra is a Hopf algebra if there exist a left and a right integral. More precisely, let A be a unital algebra (say over the field of complex numbers) with a coproduct Δ:A→A⨂A and a counit ε:A→ℂ. If there exist non-zero linear functionals φ and ψ on A satisfying (ι⨂φ)Δ(a)=φ(a)1 and (ψ⨂ι)Δ(a)=ψ(a)1 for all a∈A (where ι is the identity map on A), then there is an antipode on A and (A,Δ) is a Hopf algebra. Compare this result with the notion of a locally compact quantum group (in the von Neumann algebra setting). Given is a pair (M,Δ) of a von Neumann algebra M and a coproduct Δ:M→M⨂M (where now the von Neumann algebraic tensor product is considered). If there exist a left and a right Haar weight φ and ψ on M, then (M,Δ) is a locally compact quantum group. The key result in the theory of locally compact quantum groups is the construction of the antipode from these axioms. Then the similarity between this and the Larson-Sweedler theorem for Hopf algebras is clear. We will mainly talk about this connection. But at the end of the talk, we will briefly indicate how the same link pops up in the more recent work on quantum groupoids (joint work with B.-J. Kahng).