Schürmann, Michael; Skeide, Michael Infinitesimal generators on the quantum group \(SU_q(2)\). (English) Zbl 0924.60090 Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1, No. 4, 573-598 (1998). The theory of Lévy processes (or “white noises”) on an involutive bialgebra developed by the first author [“White noise on bialgebras” (1993; Zbl 0773.60100)] extends the classical theory of processes with independent and stationary increments on a group to the framework of quantum or non-commutative probability theory. These processes can be classified by their infinitesimal generators. In this paper the authors derive a formula for the infinitesimal generators on the quantum group \(SU_q(2)\) introduced by S. L. Woronowicz [Publ. Res. Inst. Math. Sci. 23, No. 1, 117-181 (1987; Zbl 0676.46050)]. They show that every infinitesimal generator on \(SU_q(2)\) can be decomposed into a sum of an infinitesimal generator on the one-dimensional torus and a conditionally positive functional associated with an infinite-dimensional unitary irreducible representation of \(SU_q(2)\). In particular, their result shows that the Gaussian part of a general infinitesimal generator on \(SU_q(2)\) corresponds to a Gaussian infinitesimal generator on the one-dimensional torus. Reviewer: Uwe Franz (Greifswald) Cited in 1 ReviewCited in 6 Documents MSC: 60K40 Other physical applications of random processes 81S25 Quantum stochastic calculus 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 17B37 Quantum groups (quantized enveloping algebras) and related deformations Keywords:quantum groups; Lévy processes; infinitesimal generators; Lévy-Khinchin formula Citations:Zbl 0773.60100; Zbl 0676.46050 PDF BibTeX XML Cite \textit{M. Schürmann} and \textit{M. Skeide}, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1, No. 4, 573--598 (1998; Zbl 0924.60090) Full Text: DOI OpenURL References: [1] DOI: 10.2977/prims/1195184017 · Zbl 0498.60099 [2] DOI: 10.1007/BF01162868 · Zbl 0627.60014 [3] DOI: 10.1007/BF01258530 · Zbl 0546.60058 [4] DOI: 10.1090/S0002-9947-1956-0079232-9 [5] DOI: 10.1063/1.530383 · Zbl 0785.17018 [6] Lévy P., Pisa 3 pp 337– (1934) [7] DOI: 10.1007/BF01077623 · Zbl 0679.43006 [8] DOI: 10.2977/prims/1195176848 · Zbl 0676.46050 [9] DOI: 10.1007/BF01219077 · Zbl 0627.58034 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.