The Nowicki conjecture for relatively free algebras

Author(s)

Centrone, Lucio ^[1] ; Findik, Sehmus ^[2]

Affiliation

^[1] Univ Estadual Campinas, IMECC, Rus Sergio Buarque de Holanda 651, BR-13083859 Campinas, SP - Brazil

^[2] Cukurova Univ, Dept Math, TR-01330 Adana - Turkey

• Document type: Journal article
• Source: Journal of Algebra; v. 552, p. 68-85, JUN 15 2020.
• Web of Science Citations: 0

Abstract

A linear locally nilpotent derivation of the polynomial algebra K{[}X-m] in m variables over a field K of characteristic 0 is called a Weitzenbock derivation. It is well known from the classical theorem of Weitzenbock that the algebra of constants K{[}X-m](delta) of a Weitzenbock derivation delta is finitely generated. Assume that b acts on the polynomial algebra K{[}X-2d] in 2d variables as follows: delta(x(2i)) = x(2i)(-1), delta(x(2i)(-1)) = 0, i = 1, ... , d. The Nowicki conjecture states that the algebra K{[}X-2d](delta) is generated by x(1), x(3) , X-2d(-1), and x(2i)(-)(ix2j) - x(2i)x(2j-1), 1 <= i < j <= d. The conjecture was proved by several authors based on different techniques. We apply the( )same idea to two relatively free algebras of rank 2d. We give the infinite set of generators of the algebra of constants in the free metabelian associative algebras F-2d(u), and finite set of generators in the free algebra F2d(g) in the variety determined by the identities of the infinite dimensional Grassmann algebra. (C) 2020 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 18/02108-7 - Identities in (non) associative algebras and related themes.
• Grantee: Lucio Centrone
• Support Opportunities: Regular Research Grants