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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 35, Fasc. 1,
pages 41 - 72
 

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 RP QW  N
        * , with the convolution type renormalization δn(t- s) = δ(s)δ(t- s) of the n ≥ 2 powers of the Dirac delta function, can be obtained through a contraction of the Renormalized Powers of Quantum White Noise Lie algebra RP QW  N
        c  with the scalar renormalization δn(t) = cn-1δ(t) , c > 0 . Using this renormalization, we also obtain a Lie algebra W   (c)
  ∞ which contains the w
  ∞ Lie algebra of Bakas and the Witt algebra as contractions. Motivated by the W
  ∞ algebra of Pope, Romans and Shen, we show that W   (c)
  ∞ can also be centrally extended in a non-trivial fashion. In the case of the Witt subalgebra of W
  ∞ , 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, W -algebras.

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