Rota-Baxter operators on $\ddot{I}‰$-Lie algebras

Abstract

This article explores Rota-Baxter operators on finite-dimensional $\ddot{I}‰$-Lie algebras over a field of characteristic not 2. We provide several methods for constructing left-symmetric algebras, $\ddot{I}‰$-Lie algebras, and Hom-Lie algebras via compatible Rota-Baxter operators on a given $\ddot{I}‰$-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 $\ddot{I}‰$-Lie algebra is $1$-dimensional. Furthermore, we show that for every $4$-dimensional non-Lie complex $\ddot{I}‰$-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.

21 pages; To appear in Kyushu J. Math