THE CONJECTURE OF NOWICKI ON WEITZENBÖCK DERIVATIONS OF POLYNOMIAL ALGEBRAS

Abstract

The Weitzenböck theorem states that if Δ is a linear locally nilpotent derivation of the polynomial algebra K[Z] = K[z1,…,zm] over a field K of characteristic 0, then the algebra of constants of Δ is finitely generated. If m = 2n and the Jordan normal form of Δ consists of 2 × 2 Jordan cells only, we may assume that K[Z] = K[X,Y] and Δ(yi) = xi, Δ(xi) = 0, i = 1,…,n. Nowicki conjectured that the algebra of constants K[X,Y]Δ is generated by x1,…,xn and xiyj – xjyi, 1 ≤ i < j ≤ n. Recently this conjecture was confirmed in the Ph.D. thesis of Khoury with a very computational proof, and also by Derksen whose proof is based on classical results of invariant theory. In this paper we give an elementary proof of the conjecture of Nowicki which does not use any invariant theory. Then we find a very simple system of defining relations of the algebra K[X,Y]Δ which corresponds to the reduced Gröbner basis of the related ideal with respect to a suitable admissible order, and present an explicit basis of K[X,Y]Δ as a vector space.