Details:
Title  The conjecture of Nowicki on Weitzenb\"ock derivations of polynomial algebras.  Author(s)  Vesselin Drensky, Leonid MakarLimanov  Type  Article in Journal  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.
 ISSN  02194988 
URL 
http://www.worldscientific.com/doi/abs/10.1142/S0219498809003217 
Language  English  Journal  J. Algebra Appl.  Volume  8  Number  1  Pages  4151  Publisher  World Scientific, Singapore  Year  2009  Edition  0  Translation 
No  Refereed 
No 
