A Remark About the Lie Algebra of Infinitesimal Conformal Transformations of the Euclidean Space

Infinitesimal conformal transformations of Rⁿ are always polynomialand finitely generated when n > 2. Here we prove that theLie algebra of infinitesimal conformal polynomial transformationsover Rⁿ, n 2, is maximal in the Lie algebra of polynomial vectorfields. When n is greater than 2 and p, q are such that p +q = n, this implies the maximality of an embedding of so(p +1, q + 1, R) into polynomial vector fields that was revisitedin recent works about equivariant quantizations. It also refinesa similar but weaker theorem by V. I. Ogievetsky. 1991 MathematicsSubject Classification 17B66, 53A30.