Abstract
We prove that, each probability meassure on R, with all moments, is canonically associated with (i) a *-Lie algebra; (ii) a complexity index labeled by pairs of natural integers. The measures with complexity index (0, K) consist of two disjoint classes: that of all measures with finite support and the semi-circle-arcsine class (the discussion in Sec. 4.1 motivates this name). The class C(mu) = (0, 0) coincides with the delta-measures in the finite support case and includes the semi-circle laws in the infinite support case. In the infinite support case, the class C(mu) = (0, 1) includes the arcsine laws, and the class C(mu) = (0, 2) appeared in central limit theorems of quantum random walks in the sense of Konno. The classes C(mu) = (0, K), with K >= 3, do not seem to be present in the literature. The class (1, 0) includes the Gaussian and Poisson measures and the associated *-Lie algebra is the Heisenberg algebra. The class (2, 0) includes the non-standard (i.e. neither Gaussian nor Poisson) Meixner distributions and the associated *-Lie algebra is a central extension of sl(2, R). Starting from n = 3, the *-Lie algebra associated to the class (n, 0) is infinite dimensional and the corresponding classes include the higher powers of the standard Gaussian.