Contractions and central extensions of Quantum White Noise Lie algebras

Abstrakt

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.