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Minuscule weights and random walks on lattices. (English) Zbl 0787.60089

Accardi, L. (ed.), Quantum probability and related topics. Volume VII. Singapore: World Scientific,. QP-PQ. 51-65 (1992).
Let \(G\) be a compact group and \(\varphi\) a positive type function on \(G\), continuous, with \(\varphi(e)\leq 1\). To such data a quantum generalization of the notion of random walk is associated, using the von Neumann algebra of the group \(G\). By looking at subalgebras of the von Neumann algebra of \(G\), one can recover classical stochastic processes. It is proved that if \(G\) is a semisimple compact Lie group and \(\varphi\) is the character of a representation corresponding to a minuscule weight, then two classical Markov chains obtained by looking at the subalgebra generated by a maximal torus of \(G\), and at the center of the von Neumann algebra of \(G\), are related by means of the notion of \(h\)-processes of Doob.
For the entire collection see [Zbl 0779.00006].
Reviewer: P.Biane (Paris)

MSC:

60G50 Sums of independent random variables; random walks
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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