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2010 | 89 | 1 | 13-43
Tytuł artykułu

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

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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 $w_{∞}$ 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.
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  • Centro Vito Volterra, Università di Roma Tor Vergata, via Columbia 2, 00133 Roma, Italy
  • Department of Mathematics and Natural Sciences, American College of Greece, Aghia Paraskevi 15342, Athens, Greece
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bwmeta1.element.bwnjournal-article-doi-10_4064-bc89-0-1
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