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Derivations of the Lie algebra of polynomials under Poisson bracket.


Author: L. S. Wollenberg
Journal: Proc. Amer. Math. Soc. 20 (1969), 315-320
MSC: Primary 22.90
DOI: https://doi.org/10.1090/S0002-9939-1969-0233938-1
MathSciNet review: 0233938
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Abstract: We exhibit a class of outer derivations of the Lie algebra $ P$ of complex polynomials under Poisson bracket, and prove that every derivation of $ P$ is a linear combination of one of these and an inner derivation, although this decomposition may not be unique. In particular, we show that any derivation of $ P$ which maps constants to zero must be inner. We use these results to characterise certain solutions of the Dirac problem.


References [Enhancements On Off] (What's this?)

  • [1] H. Goldstein, Classical mechanics, Addison-Wesley, Reading, Mass., 1950. MR 0043608 (13:291a)
  • [2] N. Jacobson, Lie algebras, Interscience, New York, 1962. MR 0143793 (26:1345)
  • [3] G. W. Mackey, Mathematical foundations of quantum mechanics, Benjamin, New York, 1963, Chapter 1.
  • [4] L. S. Wollenberg, Ph.D. thesis, Oxford, 1967.
  • [5] J. M. Souriau, Quantification canonique, Comm. Math. Phys. 1 (1966), 374. MR 0207332 (34:7148)
  • [6] R. F. Streater, Canonical quantization, Comm. Math. Phys. 2 (1966), 354. MR 0220490 (36:3550)
  • [7] L. van Hove, Sur certaines représentations unitaires d'un groupe infini de transformations, Acad. Roy. Belg. Cl. Sci. Mém. Coll. in 8$ ^\circ$ 26 (1951), 1. MR 0057260 (15:198d)

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DOI: https://doi.org/10.1090/S0002-9939-1969-0233938-1
Article copyright: © Copyright 1969 American Mathematical Society

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