## 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