Contractions and central extensions of quantum white noise Lie algebras

• Autorzy: Accardi L. , Boukas A.
• Języki publikacji: EN

Abstrakty

EN

We show that the Renormalized Powers of Quantum White Noise Lie algebra RPQWN_*, 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 RPQWN_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.

Słowa kluczowe

EN

contraction of a Lie algebra; renormalized powers of quantum white noise; Virasoro algebra; W-algebras

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2015
• Tom: Vol. 35, Fasc. 1
• Strony: 41--72
• Opis fizyczny: Bibliogr. 42 poz.

Twórcy

autor

Accardi L.

• Centro Vito Volterra, Università di Roma Tor Vergata, via Columbia 2, 00133 Roma, Italy

autor

Boukas A.

• Centro Vito Volterra, Università di Roma Tor Vergata, via Columbia 2, 00133 Roma, Italy

Bibliografia

• [1] L. Accardi, G. Amosov, and U. Franz, Second quantized automorphisms of the renormalized square of white noise (RSWN) algebra, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7 (2) (2004), pp. 183-194.
• [2] L. Accardi, W. Ayed, and H. Ouerdiane, White noise approach to stochastic integration, Random Oper. Stochastic Equations 13 (4) (2005), pp. 369-398.
• [3] L. Accardi, W. Ayed, and H. Ouerdiane, White noise Heisenberg evolution and Evans-Hudson flows, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (1) (2007), pp. 111-139.
• [4] L. Accardi and A. Boukas, Stochastic evolutions driven by non-linear quantum noise, Probab. Math. Statist. 22 (1) (2002), pp. 141-154.
• [5] L. Accardi and A. Boukas, Unitarity conditions for stochastic differential equations driven by nonlinear quantum noise, Random Oper. Stochastic Equations 10 (1) (2002), pp. 1-12.
• [6] L. Accardi and A. Boukas, The unitary conditions for the square of white noise, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2) (2003), pp. 197-222.
• [7] L. Accardi and A. Boukas, White noise calculus and stochastic calculus, in: Stochastic Analysis: Classical and Quantum. Perspectives of White Noise Theory – Meijo University, Nagoya, Japan, 1-5 November 2004, T. Hida (Ed.), World Scientific, 2005, pp. 260-300.
• [8] L. Accardi and A. Boukas, Higher powers of q-deformed white noise, Methods Funct. Anal. Topology 12 (3) (2006), pp. 205-219.
• [9] L. Accardi and A. Boukas, Renormalized higher powers of white noise (RHPWN) and conformal field theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (3) (2006), pp. 353-360.
• [10] L. Accardi and A. Boukas, The emergence of the Virasoro and w∞ Lie algebras through the renormalized higher powers of quantum white noise, Int. J. Math. Comput. Sci. 1 (3) (2006), pp. 315-342.
• [11] L. Accardi and A. Boukas, Powers of the delta function, QP-PQ: Quantum Probab. White Noise Anal., Vol. 20, World Scientific, 2007, pp. 33-44.
• [12] L. Accardi and A. Boukas, Fock representation of the renormalized higher powers of white noise and the centreless Virasoro (or Witt)-Zamolodchikov-w∞*-Lie algebra, J. Phys. A 41 (2008).
• [13] L. Accardi and A. Boukas, Renormalized higher powers of white noise and the Virasoro-Zamolodchikov-w∞ algebra, Rep. Math. Phys. 61 (1) (2008), pp. 1-11.
• [14] L. Accardi and A. Boukas, Analytic central extensions of infinite dimensional white noise *-Lie algebras, Stochastics 81 (3-4) (2009), pp. 201-218.
• [15] L. Accardi and A. Boukas, Cohomology of the Virasoro-Zamolodchikov and renormalized higher powers of white noise *-Lie algebras, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2) (2009), p. 1-20.
• [16] L. Accardi and A. Boukas, Quantum probability, renormalization and infinite-dimensional *-Lie algebras, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), 056.
• [17] L. Accardi and A. Boukas, The centrally extended Heisenberg algebra and its connection with Schrödinger, Galilei and renormalized higher powers of quantum white noise Lie algebra, in: Lie Theory and Its Applications in Physics – VIII International Workshop, Varna, Bulgaria, 14-21 June 2009, V. K. Dobrev (Ed.), American Institute of Physics Conference Proceedings, Vol. 1243 (2009), pp. 115-125.
• [18] L. Accardi and A. Boukas, Central extensions and stochastic processes associated with the Lie algebra of the renormalized higher powers of white noise, in: Noncommutative Harmonic Analysis with Applications to Probability, M. Bożejko, A. Krystek, and Ł. Wojakowski (Eds.), Banach Center Publ. 89 (2010), pp. 13-43.
• [19] L. Accardi, A. Boukas, and U. Franz, Renormalized powers of quantum white noise, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (1) (2006), pp. 129-147.
• [20] L. Accardi, A. Boukas, and J. Misiewicz, Existence of the Fock representation for current algebras of the Galilei algebra, in: Quantum Probability and Related Topics: Proceedings of the 30th Conference, Santiago, Chile, 23-28 November 2009, QP-PQ: Quantum Probab. White Noise Anal., Vol. 27, World Scientific, 2011, pp. 1-33.
• [21] L. Accardi and A. Dhahri, The quadratic Fock functor, J. Math. Phys. 51 (2010), 022105.
• [22] L. Accardi and A. Dhahri, C*-non-linear second quantization, arXiv: 1401.5500v2, September 12, 2014.
• [23] L. Accardi, U. Franz, and M. Skeide, Squared white noise and other non-Gaussian noises as Lévy processes on real Lie algebras, Comm. Math. Phys. 228 (2002), pp. 123-150.
• [24] L. Accardi, Y. G. Lu, and N. Obata, Towards a non-linear extension of stochastic calculus, RIMS Kokyuroku 957 (1996), pp. 1-15.
• [25] L. Accardi, Y. G. Lu, and I. V. Volovich, White noise approach to classical and quantum stochastic calculi, Lecture Notes of the Volterra International School of the same title, Trento, Italy, 1999; Volterra Center preprint No. 375, Università di Roma Tor Vergata, July 1999.
• [26] L. Accardi, H. Ouerdiane, and H. Rebei, On the quadratic Heisenberg group, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 (4) (2010), pp. 551-587.
• [27] L. Accardi and I. V. Volovich, Quantum white noise with singular non-linear interaction (Development of Infinite-Dimensional Noncommutative Analysis), N. Obata (Ed.), RIMS Kokyuroku 1099 (1999), pp. 61-69.
• [28] I. Bakas, The structure of the Winf algebra, Comm. Math. Phys. 134 (1990), pp. 487-508.
• [29] A. Barhoumi and A. Riahi, Nuclear realization of the Virasoro-Zamolodchikov-w∞ *-Lie algebras through the renormalized higher powers of the quantum Meixner white noise, Open Syst. Inf. Dyn. 17 (2) (2010), pp. 135-160.
• [30] F. A. Berezin, The Method of Second Quantization, Academic Press, New York 1966.
• [31] M. Bożejko and W. Bryc, A quadratic regression problem for two-state algebras with application to the Central Limit Theorem, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2) (2009), pp. 231-249.
• [32] P. J. Feinsilver and R. Schott, Algebraic Structures and Operator Calculus. Volumes I and III, Kluwer, 1993.
• [33] B. Grigelionis, Processes of Meixner type, Lithuanian Math. J. 39 (1999), pp. 33-41.
• [34] B. Grigelionis, Generalized z-distributions and related stochastic processes, Lithuanian Math. J. 41 (2001), pp. 303-319.
• [35] T. Hida, Selected Papers of Takeyuki Hida, L. Accardi et al. (Eds.), World Scientific, 2001.
• [36] S. V. Ketov, Conformal Field Theory, World Scientific, 1995.
• [37] C. N. Pope, Lectures on W algebras and W gravity, Lectures given at the Trieste Summer School in High-Energy Physics, August 1991.
• [38] C. N. Pope, L. J. Romans, and X. Shen,Winf and the Racah-Wigner algebra, Nuclear Phys. B 339 (1990), pp. 191-221.
• [39] I. E. Segal, Transformations in Wiener space and squares of quantum fields, Adv. Math. 4 (1970), pp. 91-108.
• [40] P. Śniady, Quadratic bosonic and free white noises, Comm. Math. Phys. 211 (3) (2000), pp. 615-628.
• [41] A. B. Zamolodchikov, Infinite extra symmetries in two-dimensional conformal quantum field theory (in Russian), Teoret. Mat. Fiz. 65 (3) (1985), pp. 347-359.
• [42] L. Accardi and A. Boukas, The Schrödinger Fock kernel and the no-go theorem for the first order and Renormalized Square of White Noise Lie algebras, Proc. Amer. Math. Soc. 139 (8) (2011), pp. 2973-2986. [The reference added in proof.].

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).