Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T23:27:50.364Z Has data issue: false hasContentIssue false
  • English
  • Français

Chaos map for the universal enveloping algebra of U(N)*

Published online by Cambridge University Press:  24 October 2008

R. L. Hudson  and
K. R. Parthasarathy
R. L. Hudson
Affiliation:
Mathematics Department, University of Nottingham, Nottingham NG7 2RD
K. R. Parthasarathy
Affiliation:
Mathematics Department, University of Nottingham, Nottingham NG7 2RD
Get access Rights & Permissions [Opens in a new window]

Abstract

It is shown that the family of representations {jt, t ∈ ℝ+} of the universal enveloping algebra U of the N-dimensional unitary group which is generated by the N-dimensional number process of quantum stochastic calculus can be expressed in the form

where ψ is a bijective linear map from U onto the space S of symmetric tensors over the Lie algebra, and It is the iterated (chaotic) integral on S. The chaotic product * is defined by the formula

and satisfies

This work generalizes and completes earlier results on the centre of U.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 1 , January 1995 , pp. 21 - 30
Copyright
Copyright © Cambridge Philosophical Society 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[Gro]Gross, L.. The homogeneous chaos over compact Lie groups, pp. 117–123. In Stochastic Processes, a Festschrift in honour of Gopinath Kallianpur, ed. Cambanis, S. et al. (Springer, 1993).CrossRefGoogle Scholar
[Hud]Hudson, R. L.. A new system of Casimir operators for U(n), Journal of Mathematical Physics 15 (1974), 1067–1070.CrossRefGoogle Scholar
[HP1]Hudson, R. L. and Parthasarathy, K. R.. Quantum Itô's formula and stochastic evolutions. Commun. Math. Phys. 93 (1984) 301–323.CrossRefGoogle Scholar
[HP2]Hudson, R. L. and Parthasarathy, K. R.. The Casimir chaos map for U(N). Tatra Mountains Math Publ. 3 (1993), 81–88.Google Scholar
[HP3]Hudson, R. L. and Parthasarathy, K. R.. Casimir chaos in Boson Fock space. J. Funct. Anal. 119 (1984) 319–339.CrossRefGoogle Scholar
[Par]Parthasarathy, K. R.. An introduction to quantum stochastic calculus (Birkhäuser, 1992).Google Scholar