Definition 0.1 A locally compact quantum group defined as in ref. [1] is a quadruple
QCGl = (A, Δ,μ,ν), where A is either a C∗- or a W∗ - algebra equipped with a co-associative
comultiplication Δ : A → A ⊗ A and two faithful semi-finite normal weights, μ and ν - right and
-left Haar measures.
Examples
-
1.
- An ordinary unimodular group G with Haar measure μ: A = L∞(G,μ), Δ :
f(g)
f(gh), S : f(g)
f(g−1),ϕ(f) = ∫
Gf(g)dμ(g), where g,h ∈ G,f ∈ L∞(G,μ).
-
2.
- A = Ł(G) is the von Neumann algebra generated by left-translations Lg or by left
convolutions Lf = ∫
G(f(g)Lgdμ(g)) with continuous functions f() ∈ L1(G,μ)Δ :
Lg ⊗ Lg
Lg−1,ϕ(f) = f(e), where g ∈ G, and e is the unit of G.
References
[1] Leonid Vainerman. 2003.Locally Compact Quantum Groups and Groupoids:
Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg,
February 21-23, 2002 Series in Mathematics and Theoretical Physics, 2, Series ed. V.
Turaev., Walter de Gruyter Gmbh et Co: Berlin.
