Polynomial Lie Algebras

Following the work of Buchstaber and Leikin we will study infinite-dimensional Lie algebras which are modules of finite rank over the ring of polynomials. Such Lie algebras are known as polynomial Lie algebras. Every polynomial Lie algebra is defined by polynomial structural constants. Many examples of such algebras will be given. The main purpose of the paper is to prove that vector fields defining the canonical representation of a polynomial Lie algebra are tangent to the hyperplane $\mathrm{det}(V) = 0$, where $V$ is the matrix of structural constants.