Abstract
Let W n (ℝ) be the Weyl algebra of index n. It is well known that so(p, q) Lie algebras can be viewed as quadratic polynomial (Lie) algebras in W n (ℝ) for p + q = n with the Lie algebra multiplication being given by the bracket [a, b] = ab − ba for a, b quadratic polynomials in W n (ℝ). What does not seem to be so well known is that the converse statement is, in a certain sense, also true, namely, that, by using extension and localization, it is possible, at least in some cases, to construct homomorphisms of W n (ℝ) onto its image in a localization of U(so(p + 2, q)), the universal enveloping algebra of so(p + 2, q), and m = p + q. Since Weyl algebras are simple, these homomorphisms must either be trivial or isomorphisms onto their images. We illustrate this remark for the so(2, q) case and construct a mappping from W q (ℝ) onto its image in a localization of U(so(2, q)). We prove that this mapping is a homomorphism when q = 1 or q = 2. Some specific results about representations for the lowest dimensional case of W 1(ℝ) and U(so(2, 1)) are given.
Similar content being viewed by others
References
I. M. Gelfand and A. A. Kirillov, Dokl. Akad. Nauk SSSR 167, 503 (1966) [Sov. Math. 7, 403 (1966)].
A. Premet, Invent. Math. 181, 395 (2010).
David A. Vogan, Unitary Representations of Reductive Lie Groups, Annals of Mathematics Studies (Princeton Univ. Press, Princeton, 1987), p. 118.
S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York, 1962).
Jacques Dixmier, Enveloping Algebras (North-Holland, Amsterdam, 1977).
P. Moylan, Studies in Mathematical Physics Research, (Nova Science, New York, 2004).
V. K. Dobrev, G. Mack, V. B. Petkova, et al., Harmonic Analysis on the n-Dimensional Lorentz Group and Its Applications to Conformal Quantum Field Theory (Springer-Verlag, Berlin, 1977), Lect. Not. Phys. 63.
V. K. Dobrev, J. Phys. A 39, 5995 (2006).
Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples (Princeton Univ. Press, Princeton, 1986).
J. Sekiguchi, Nagoya Math. J. 79, 151 (1980).
B. Ørsted and I. E. Segal, J. Funct. Anal. 83, 150 (1989).
I. E. Segal, Duke Math. J. 18, 2212 (1951).
E. Inönü and E. P. Wigner, Proc. Natl. Acad. Sci. 39, 510 (1953).
Author information
Authors and Affiliations
Corresponding author
Additional information
The text was submitted by the author in English.
Rights and permissions
About this article
Cite this article
Moylan, P. Localization and the Weyl algebras. Phys. Atom. Nuclei 80, 590–597 (2017). https://doi.org/10.1134/S106377881703022X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S106377881703022X