Rota-Baxter operators on $\omega$-Lie algebras

TL;DR

This paper investigates Rota-Baxter operators on finite-dimensional $ ext{ω}$-Lie algebras, providing methods for constructing related algebraic structures and analyzing their geometric properties, especially over complex numbers.

Contribution

It introduces new methods for constructing compatible Rota-Baxter operators and studies their geometric structures on $ ext{ω}$-Lie algebras, including classification results.

Findings

The variety of isometric Rota-Baxter operators of weight 1 is 1-dimensional on certain $ ext{ω}$-Lie algebras.

Existence of nilpotent compatible Rota-Baxter operators on 4-dimensional $ ext{ω}$-Lie algebras.

Induced Hom-Lie algebras can be nonabelian but solvable.

Abstract

This article explores Rota-Baxter operators on finite-dimensional $\omega$-Lie algebras over a field of characteristic not 2. We provide several methods for constructing left-symmetric algebras, $\omega$-Lie algebras, and Hom-Lie algebras via compatible Rota-Baxter operators on a given $\omega$-Lie algebra. We also study the geometric structures of compatible Rota-Baxter operators of weight $0$ and isometric Rota-Baxter operators of weight $1$ over the field of complex numbers. In particular, we prove that the affine variety of all isometric Rota-Baxter operators of weight $1$ on any finite-dimensional non-Lie complex simple $\omega$-Lie algebra is $1$-dimensional. Furthermore, we show that for every $4$-dimensional non-Lie complex $\omega$-Lie algebra, there always exists a nilpotent compatible Rota-Baxter operator of weight $0$ such that the induced Hom-Lie algebra is nonabelian but solvable.