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Infinitesimal generators on the quantum group \(SU_q(2)\). (English) Zbl 0924.60090

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.

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
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References:

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