Abstract
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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