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Further Applications to Lie Algebras

Part of the Frontiers in Mathematics book series (FM)

Abstract

In this closing chapter we consider three rather unrelated applications of FI’s. The common property of all three topics is the Lie algebra framework. But otherwise, each of them has a different background. We shall discuss the motivation and history just briefly (mostly at the end of the chapter), and in each section we will introduce the necessary notions in a very concise manner. It is not our intention to go into the heart of the matter of these topics. Our main goal is to indicate that FI’s are hidden behind various mathematical notions, and after tracing them out one can effectively apply the theory presented in Part II.

Keywords

Associative Algebra Poisson Algebra Prime Algebra Nonassociative Algebra Witt Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Literature and Comments

  1. [2]
    A. A. Albert, Power-associative rings, Trans. Amer. Math. Soc. 64 (1948), 318–328.CrossRefMathSciNetGoogle Scholar
  2. [28]
    K. I. Beidar, M.A. Chebotar, On Lie-admissible algebras whose commutator Lie algebras are Lie subalgebras of prime associative algebras, J. Algebra 233 (2000), 675–703.MATHCrossRefMathSciNetGoogle Scholar
  3. [33]
    K. I. Beidar, M.A. Chebotar, Y. Fong, W.-F. Ke, On some Lie-admissible subalgebras of matrix algebras, J. Math. Sci. 131 (2005), 5939–5947.CrossRefMATHMathSciNetGoogle Scholar
  4. [44]
    G. M. Benkart, Power-associative Lie-admissible algebras, J. Algebra 90 (1984), 37–58.MATHCrossRefMathSciNetGoogle Scholar
  5. [45]
    G. M. Benkart, J. M. Osborn, Flexible Lie-admissible algebras, J. Algebra 71 (1981), 11–31.MATHCrossRefMathSciNetGoogle Scholar
  6. [46]
    G. M. Benkart, J.M. Osborn, Power-associative products on matrices, Hadronic J. Math. 5 (1982), 1859–1892.MATHMathSciNetGoogle Scholar
  7. [93]
    M. A. Chebotar, W.-F. Ke, On skew-symmetric maps on Lie algebras. Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), 1273–1281.MATHCrossRefMathSciNetGoogle Scholar
  8. [105]
    D. R. Farkas, Characterization of Poisson algebras, Comm. Algebra 23 (1995), 4669–4686.MATHCrossRefMathSciNetGoogle Scholar
  9. [106]
    D. R. Farkas, Poisson polynomial identities, Comm. Algebra 26 (1998), 401–416.MATHCrossRefMathSciNetGoogle Scholar
  10. [107]
    D. R. Farkas, G. Letzter, Ring theory from symplectic geometry, J. Pure Appl. Algebra 125 (1998), 155–190.MATHCrossRefMathSciNetGoogle Scholar
  11. [123]
    K. Jeong, S.-J. Kang, H. Lee, Lie-admissible algebras and Kac-Moody algebras, J. Algebra 197 (1997), 492–505.MATHCrossRefMathSciNetGoogle Scholar
  12. [125]
    A. Joseph, Derivations of Lie brackets and canonical quantisation, Comm. Math. Phys. 17 (1970), 210–232. p 219MATHCrossRefMathSciNetGoogle Scholar
  13. [127]
    W.-F. Ke, M.A. Chebotar, On biadditive mappings of some Lie algebras, Russian Math. Surveys 58 (2003), 183–184.MATHCrossRefMathSciNetGoogle Scholar
  14. [129]
    F. Kubo, Finite-dimensional non-commutative Poisson algebras, J. Pure Appl. Algebra 113 (1996), 307–314.MATHCrossRefMathSciNetGoogle Scholar
  15. [130]
    F. Kubo, Non-commutative Poisson algebra structures on affine Kac-Moody algebras, J. Pure Appl. Algebra 126 (1998), 267–280.MATHCrossRefMathSciNetGoogle Scholar
  16. [136]
    F. J. Laufer and M. L. Tomber, Some Lie-admissible algebras, Canad. J. Math. 14 (1962), 287–292.MATHMathSciNetGoogle Scholar
  17. [171]
    H. C. Myung, Some classes of flexible Lie-admissible algebras, Trans. Amer. Math. Soc. 167 (1972), 79–88.MATHCrossRefMathSciNetGoogle Scholar
  18. [172]
    H. C. Myung, Malcev-admissible algebras, “Progress in Mathematics”, Vol. 64, Birkhäuser, 1986.Google Scholar
  19. [173]
    Y. Nambu, Generalized Hamiltonian dynamics, Phys. Rev. D. 7 (1973), 2405–2412.CrossRefMathSciNetGoogle Scholar
  20. [174]
    S. Okubo, Introduction to octonion and other non-associative algebras in physics, Cambridge University Press, New York, 1995. Chapter 8MATHGoogle Scholar
  21. [175]
    S. Okubo, H.C. Myung, Adjoint operators in Lie algebras and the classification of simple flexible Lie-admissible algebras, Trans. Amer. Math. Soc. 264 (1981), 459–472.MATHCrossRefMathSciNetGoogle Scholar
  22. [188]
    R. M. Santilli, Embedding of a Lie algebra in nonassociative structures, Nuovo Cimento A 51 (1967), 570–576.MATHCrossRefMathSciNetGoogle Scholar
  23. [190]
    J. M. Souriau, Quantification géométrique, Comm. Math. Phys. 1 (1965/1966), 374–398.MathSciNetGoogle Scholar
  24. [191]
    R. F. Streater, Canonical quantization, Comm. Math. Phys. 2 (1966), 354–374.MATHCrossRefMathSciNetGoogle Scholar

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© Birkhäuser Verlag 2007

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