The image of Lie polynomials on real Lie algebras of dimension up to 3


Centrone L., Fındık Ş.

Journal of Algebra, vol.659, pp.344-360, 2024 (SCI-Expanded, Scopus)

  • Publication Type: Article / Article
  • Volume: 659
  • Publication Date: 2024
  • Doi Number: 10.1016/j.jalgebra.2024.07.006
  • Journal Name: Journal of Algebra
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Page Numbers: pp.344-360
  • Keywords: Images of polynomials, Lie algebras
  • Çukurova University Affiliated: No

Abstract

Let Fn be the free Lie algebra over R of rank n generated by y1,…,yn, and let f∈Fn′ be a multilinear Lie polynomial contained in the commutator ideal Fn′ of Fn. In this paper, we determine the image Imf={f(w1,…,wn)|wi∈L,i=1,…,n}⊂L, for Lie algebras L of dimension ≤3, and of the Lie algebra of dimension 4 stated in a paper of Baker dating back to 1901. In all the cases studied, the L'vov-Kaplansky Conjecture has a positive answer.