Contractions and central extensions of Quantum White Noise Lie algebras
Słowa kluczowe:Contraction of a Lie algebra, renormalized powers of quantum white noise, Virasoro algebra, W-algebras
We show that the Renormalized Powers of Quantum White Noise Lie algebra RPQWN, with the convolution type renormalization δnt − s = δsδt − 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 RPQWNc with the scalar renormalization δnt = cn−1δt, 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.