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

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.