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Polynomial realizations and derivations of Poisson-Bracket Lie subalgebras

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Il Nuovo Cimento B (1971-1996)

Summary

We study the isomorphic polynomial realizations of abstract Lie algebras as a subalgebra ℛ of the Poisson-bracket Lie algebra of all polynomials ℒ, supposing mostly that ℛ is generated by monomials. The problem is to describe the outer derivations of ℛ as induced by some derivations of the ambient Lie algebra ℒ (called here Wollenberg-type derivations) and some inner derivations of another ambient Lie algebra Q which are eventually a non-polynomial Lie-algebra extension of the given ℛ. Here we describe the solution in the case of a finite-generated Lie algebra ℛ. Explicit results are obtained for some 3-generated polynomial Lie subalgebras. As an application we obtain some relations of constrained, especially constrained submanifolds of Heisenberg type and constrained derivation pairs of subalgebras.

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Authors and Affiliations

  1. Department of Mathematics, Higher Institute of Architecture in Construction, Bul. Ch. Smirnenski No. 1, 1421, Sofia, Bulgaria

    E. J. Arnaudova

  2. Mathematical Institute of the Bulgarian Academy of Sciences, P.O. Box 373, 1090, Sofia, Bulgaria

    S. G. Dimiev

  3. Department of Mathematics Panemistemiopolis, University of Athens, 15784, Athens, Greece

    L. Ch. Papaloucas

  4. Department of Mathematics, Higher Institute of Mathematics, Bul. Ch. Smimeuski No. 1, 1421, Sofia, Bulgaria

    I. I. Tatarova

Authors
  1. E. J. Arnaudova
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  2. S. G. Dimiev
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  3. L. Ch. Papaloucas
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  4. I. I. Tatarova
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Arnaudova, E.J., Dimiev, S.G., Papaloucas, L.C. et al. Polynomial realizations and derivations of Poisson-Bracket Lie subalgebras. Nuov Cim B 108, 1131–1144 (1993). https://doi.org/10.1007/BF02827309

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  • DOI: https://doi.org/10.1007/BF02827309

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