CONTRACTIONS AND CENTRAL EXTENSIONS OF QUANTUM WHITE
NOISE LIE ALGEBRAS

Luigi Accardi

Andreas Boukas

Abstract: We show that the Renormalized Powers of Quantum White Noise Lie algebra
, with the convolution type renormalization of the
powers of the Dirac delta function, can be obtained through a contraction of the
Renormalized Powers of Quantum White Noise Lie algebra with the scalar
renormalization , . Using this renormalization, we also obtain a Lie
algebra which contains the Lie algebra of Bakas and the Witt algebra as
contractions. Motivated by the algebra of Pope, Romans and Shen, we show that
can also be centrally extended in a non-trivial fashion. In the case of the Witt
subalgebra of , the central extension coincides with that of the Virasoro algebra.

2000 AMS Mathematics Subject Classification: Primary: 60B15; Secondary:
60H40.

Keywords and phrases: Contraction of a Lie algebra, renormalized powers of quantum
white noise, Virasoro algebra, -algebras.