POLYNOMIAL REALIZATIONS AND DERIVATIONS OF POISSON-BRACKET LIE SUBALGEBRAS

We study the isomorphic polynomial realizations of abstract Lie algebr as as a subalgebra R of the Poisson-bracket Lie algebra of all polynom ials L, supposing mostly that R is generated by monomials. The problem is to describe the outer derivations of R as induced by some derivati ons of the ambient Lie algebra L (called here Wollenberg-type derivati ons) 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 a lgebra R. Explicit results are obtained for some 3-generated polynomia l Lie subalgebras. As an application we obtain some relations of const rained, especially constrained submanifolds of Heisenberg type and con strained derivation pairs of subalgebras.