Lie nilpotent Novikov algebras and Lie solvable Leavitt path algebras

TL;DR

This paper investigates the properties of Novikov algebras and characterizes when Leavitt path algebras are Lie solvable or Lie nilpotent, providing a complete classification in these cases.

Contribution

It establishes conditions for Lie solvability of Leavitt path algebras and describes Lie nilpotent Leavitt path algebras completely, linking Lie solvability to Lie nilpotency.

Findings

The ideal generated by commutators in Lie nilpotent Novikov algebras is nilpotent.

Necessary and sufficient conditions for Lie solvability of Leavitt path algebras are provided.

Lie solvability of Leavitt path algebras coincides with the Lie nilpotency of their commutator subalgebra.

Abstract

In this paper, we first study properties of the lower central chains for Novikov algebras. Then we show that for every Lie nilpotent Novikov algebra~$\mathcal{N}$, the ideal of~$\mathcal{N}$ generated by the set~$\{ab - ba\mid a, b\in \mathcal{N}\}$ is nilpotent. We secondly provide necessary and sufficient conditions on the graph $E$ and the field $K$ for which the Leavitt path algebra $L_K(E)$ is Lie solvable. Consequently, we obtain a complete description of Lie nilpotent Leavitt path algebras, and show that the Lie solvability of~$L_K(E)$ and the Lie nilpotency of $[L_K(E),L_K(E)]$ are the same.