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Lévy–Khintchine decompositions for generating functionals on algebras associated to universal compact quantum groups

    https://doi.org/10.1142/S0219025718500170Cited by:4 (Source: Crossref)

    We study the first and second cohomology groups of the -algebras of the universal unitary and orthogonal quantum groups U+F and O+F. This provides valuable information for constructing and classifying Lévy processes on these quantum groups, as pointed out by Schürmann. In the case when all eigenvalues of FF are distinct, we show that these -algebras have the properties (GC), (NC) and (LK) introduced by Schürmann and studied recently by Franz, Gerhold and Thom. In the degenerate case F=Id, we show that they do not have any of these properties. We also compute the second cohomology group of U+d with trivial coefficients — H2(U+d,𝜖𝜖)d21 — and construct an explicit basis for the corresponding second cohomology group for O+d (whose dimension was known earlier, thanks to the work of Collins, Härtel and Thom).

    Communicated by M. Bozejko

    AMSC: Primary: 16T20, Secondary: 16T05
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