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 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc89-0-1,
author = {Luigi Accardi and Andreas Boukas},
title = {Central extensions and stochastic processes associated with the Lie algebra of the renormalized higher powers of white noise},
journal = {Banach Center Publications},
volume = {89},
year = {2010},
pages = {13-43},
zbl = {1214.60032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc89-0-1}
}
Luigi Accardi; Andreas Boukas. Central extensions and stochastic processes associated with the Lie algebra of the renormalized higher powers of white noise. Banach Center Publications, Tome 89 (2010) pp. 13-43. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc89-0-1/