Lie algebras canonically associated to probability measures on R with all moments

In the paper Accardi et al.: Identification of the theory of orthogonal polynomials in d–indeterminates with the theory of 3–diagonal symmetric interacting Fock spaces on Cd, submitted to: IDA–QP (Infinite Dimensional Anal. Quantum Probab. Related Topics), [1], it has been shown that, with the natural definitions of morphisms and isomorphisms (that will not be recalled here) the category of orthogonal polynomials in a finite number of variables is isomorphic to the category of symmetric interacting fock spaces (IFS) with a 3–diagonal structure. Any IFS is canonically associated to a ∗–Lie algebra (commutation relations) and a ∗–Jordan algebra (anti–commutation relations). In this paper we continue the study of these algebras, initiated in Accardi et al. An Information Complexity index for Probability Measures on R with allmoments, submitted to: IDA–QP (Infinite Dimensional Anal. Quantum Probab. Related Topics), [2], in the case of polynomials in one variable, refine the definition of information complexity index of a probability measure on the real line, introduced there, and prove that the ∗–Lie algebra canonically associated to the probability measures of complexity index (0, K, 1), defining finite– dimensional approximations, in the sense of Jacobi sequences, of the Heisenberg algebra, coincides with the algebra of all K × K complex matrices.