Integrable sigma model with generalized $\mathcal{F}$ structure, Yang-Baxter sigma model with generalized complex structure and multi-Yang-Baxter sigma model

We construct an integrable sigma model with a generalized $\mathcal{F}$ structure, which involves a generalized Nijenhuis structure $\mathcal{J}$ satisfying $\mathcal{J}^{3}=-\mathcal{J}$. Utilizing the expression of the generalized complex structure on the metric Lie group manifold $G$ in terms of operator relations on its Lie algebra $\mathfrak{g}$, we formulate a Yang-Baxter sigma model with a generalized complex structure. Additionally, we present multi-Yang-Baxter sigma models featuring two and three compatible Nijenhuis structures. Examples for each of these models are provided.