Lie algebras associated with the renormalized higher powers of white noise

We recall the recently established (cf. [1] and [2]) connection between the renormalized higher powers of white noise (RHPWN) *-Lie algebra and the Virasoro--Zamolodchikov$w_{\infty}*$--Lie algebra of conformal field theory (cf. [10]). Motivated by this connection, with the goal of investigating a possible connection with classical independent increments processes, we begin a systematic study of the sub-*{Lie algebras of the (1{mode) full oscillator algebra. This program has two additional motivations: (i) the full oscillator algebra is a fundamental object of mathematics and the structure of its subalgebras deserves deep investigation; (ii) the no--go theorems show that the current algebras over some Lie sub-- algebras of Lie algebras may have a Fock representation individually without this being true for the Lie algebra generated by them. The problem of classifying which sub--algebras of the full oscillator algebra have this property is open and a preliminary step towards its analysis is the classification of the ''natural" sub algebras of the full oscillator algebra. We construct two hierarchies of such sub algebras, parametrized by the natural integers. One of these hierarchies begins with the Virasoro algebra. Another possibility to bypass the no--go theorems is to consider different renormalizations of the higher powers of white noise commutation relations. This approach is developed in Section 2, where we show with examples that some of them lead to known (i.e., first or second order) commutation relations. This fact is probably related with the gaussianization phenomenon discussed in [7].