The Nowicki conjecture for relatively free algebras

Abstract

A linear locally nilpotent derivation of the polynomial algebra K[Xm] in m variables, over a field K of characteristic 0, is called a Weitzenböck derivation. It is well known from the classical theorem of Weitzenböck that the algebra of constants K[Xm] δ of a Weitzenböck derivation δ is finitely generated. Let m = 2d, and the Weitzenböck derivation δ act on the the polynomial algebra K[X 2d ] in 2d variables as follows:The conjecture was proved by several authors based on different techniques. We apply the same idea to several relatively free algebras of rank 2d. We give the infinite set of generators of the algebra of constants in the the free metabelian associative algebras F 2d (A), and finite set of generators in the free centre-by-metabelian Lie algebras F 2d (C) and the free algebra F 2d (G) in the variety generated by the identities of infinite dimensional Grassmann algebra.2010 Mathematics Subject Classification. 17B01; 17B30; 17B40; 16R10; 16S15;16W25; 13N15; 13A50.Key words and phrases. Free metabelian associative algebras; free centre-by-metabelian Lie algebras; algebras of constants; Weitzenböck derivations, the Nowicki conjecture.

On Nowicki's conjecture: a survey and a new result

“…The environment now is that of relatively free algebras. The complete proofs of the results presented in the next two section can be found in the paper [4].…”

“…Here δ acts on U d and V d in the same way as on X d . In [4] the authors give an explicit set of generators of the algebra of constants of the variety generated by the infinite dimensional Grassmann algebra and of the free metabelian associative algebra. A good part of those result is presented explicitly in the text (Sections 5 and 6).…”

The Nowicki Conjecture for Bicommutative Algebras

“…They gave a finite generating set for the algebra (𝐹 2𝑛 ′ ) 𝛿 included in the commutator ideal 𝐹 2𝑛 ′ of 𝐹 2𝑛 as a 𝐾[𝑋 𝑛 , 𝑌 𝑛 ] 𝛿 -module. As a continuation of this work a finite generation set for the algebra of constants in the commutator ideal of the free metabelian associative algebra generated by 𝑋 𝑛 ∪ 𝑌 𝑛 as a 𝐾[𝑋 𝑛 , 𝑌 𝑛 ] 𝛿 -bimodule was given in [11]. In the same work, a set of finite generators was obtained for the free algebra in the variety of infinite dimensional Grassmann algebras.…”

İzi̇ Sifir Olan Jeneri̇k Matri̇sler Cebi̇ri̇ İçi̇n Nowicki Sanisi

“…One may count the Nowicki conjecture for the free metabelian Lie algebra 𝐹 2𝑑 of rank 2𝑑 [10], in which a finite generating set for the algebra (𝐹 2𝑑 ′ ) 𝛿 as a 𝐾[𝑋 𝑑 , 𝑌 𝑑 ] 𝛿 -module was given. Additinally, the Nowicki conjecture was studied for the free metabelian associative algebra of rank 2𝑑 [11]. Also, generators were obtained for algebras of invariants in Grassmann algebras [11].…”

“…Additinally, the Nowicki conjecture was studied for the free metabelian associative algebra of rank 2𝑑 [11]. Also, generators were obtained for algebras of invariants in Grassmann algebras [11]. Finally, the free metabelian Possion algebra was considered in [12].…”