Weitzenbock derivations of free metabelian associative algebras

WOS: 000398858300001By the classical theorem of Weitzenbock the algebra of constants K[X-d](delta) of a nonzero locally nilpotent linear derivation d of the polynomial algebra K[X-d] = K[x(1),..., x(d)] in several variables over a field K of characteristic 0 is finitely generated. As a noncommutative generalization one considers the algebra of constants F-d(V)(delta) of a locally nilpotent linear derivation d of a finitely generated relatively free algebra F-d(V)(delta) in a variety Vof unitary associative algebras over K. It is known that F-d(V)(delta) is finitely generated if and only if Vsatisfies a polynomial identity which does not hold for the algebra U-2(K) of 2x2 upper triangular matrices. Hence the free metabelian associative algebra F-d = F-d(M) = F-d(N(2)A) = F-d(var(U-2(K))) is a crucial object to study. We show that the vector space of the constants (F'(d))(delta) in the commutator ideal F'd is a finitely generated K[U-d, V-d](delta)-module, where d acts on Ud and Vd in the same way as on Xd. For small d, we calculate the Hilbert series of(F'(d))(delta) and find the generators of the K[U-d, V-d](delta)- module (F'(d))(delta). This gives also an (infinite) set of generators of the algebra F-d(delta).Bulgarian Science Fund for Bilateral Scientific Cooperation between Bulgaria and Ukraine [Ukraine 01/0007]; Council of Higher Education (YOK) in TurkeyMinistry of National Education - Turkey; Bulgarian National Science FundNational Science Fund of Bulgaria [I02/18]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. 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. The research of the third-named author was partially supported by the Council of Higher Education (YOK) in Turkey during his visit as a post-doctoral fellow at the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences. He is very thankful to the Institute for the creative atmosphere and the warm hospitality for the period when this project was carried out. For the revised version of the paper, the research of the second- and the third-named authors was partially supported by Grant I02/18 of the Bulgarian National Science Fund