Weitzenböck derivations of free metabelian Lie algebras

A nonzero locally nilpotent linear derivation ? of the polynomial algebra K[Xd]=K[x1,.,Xd] in several variables over a field K of characteristic 0 is called a Weitzenböck derivation. The classical theorem of Weitzenböck states that the algebra of constants K[Xd]? (which coincides with the algebra of invariants of a single unipotent transformation) is finitely generated. Similarly one may consider the algebra of constants of a locally nilpotent linear derivation ? of a finitely generated (not necessarily commutative or associative) algebra which is relatively free in a variety of algebras over K. Now the algebra of constants is usually not finitely generated. Except for some trivial cases this holds for the algebra of constants (Ld/Ld ¨)? of the free metabelian Lie algebra L d/Ld¨ with d generators. We show that the vector space of the constants (Ld/Ld¨) ? in the commutator ideal Ld'/L d¨ is a finitely generated K[Xd]?-module. For small d, we calculate the Hilbert series of (Ld/Ld ¨)? and find the generators of the K[X d]?-module (Ld/Ld¨) ?. This gives also an (infinite) set of generators of the algebra (Ld/Ld¨)?. © 2013 Elsevier Inc. All rights reserved.Bulgarian National Science Fund Council for Higher EducationCorresponding author. Tel.: +359 2 9792820; fax: +359 2 9713649. E-mail addresses: dangovski@gmail.com (R. Dangovski), drensky@math.bas.bg (V. Drensky), sfindik@cu.edu.tr (S¸ . Fındık). 1 The research of the first named author was a part of his project in the frames of the High School Student Institute at the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences. 2 The research of the second named author was partially supported by Grant Ukraine 01/0007 of the Bulgarian Science Fund for Bilateral Scientific Cooperation between Bulgaria and Ukraine. 3 The research of the third named author was partially supported by the Council of Higher Education (YÖK) in Turkey