## Central extensions and stochastic processes associated with the Lie algebra of the renormalized higher powers of white noise

### Volume 89 / 2010

#### Abstract

In the first part of the paper we discuss possible definitions of Fock representation of the $*$-Lie algebra of the Renormalized Higher Powers of White Noise ($RHPWN$). We propose one definition that avoids the no-go theorems and we show that the vacuum distribution of the analogue of the field operator for the $n$-th renormalized power of WN defines a continuous binomial process. In the second part of the paper we present without proof our recent results on the central extensions of $RHPWN$, its subalgebras and the $w_{\infty}$ Lie algebra of conformal field theory. In the third part of the paper we describe our results on the non-trivial central extensions of the Heisenberg algebra. This is a 4-dimensional Lie algebra, hence belonging to a list which is well known and has been studied by several research groups. However the canonical nature of this algebra, i.e. the fact that it is the unique (up to a complex scaling) non-trivial central extension of the Heisenberg algebra, seems to be new. We also find the possible vacuum distributions corresponding to a family of injective $*$-homomorphisms of different non-trivial central extensions of the Heisenberg algebra into the Schrödinger algebra.