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F. Labourie [arXiv:1212.5015] characterized the Hitchin components for$\operatorname{PSL}(n, \mathbb{R})$ for any $n>1$ by using the swappingalgebra, where the swapping algebra should be understood as a ring equippedwith a Poisson bracket. We introduce the rank $n$ swapping algebra, which isthe quotient of the swapping algebra by the $(n+1)\times(n+1)$ determinantrelations. The main results are the well-definedness of the rank $n$ swappingalgebra and the "cross-ratio" in its fraction algebra. As a consequence, we usethe sub fraction algebra of the rank $n$ swapping algebra generated by these"cross-ratios" to characterize the $\operatorname{PSL}(n, \mathbb{R})$ Hitchincomponent for a fixed $n>1$. We also show the relation between the rank $2$swapping algebra and the cluster$\mathcal{X}_{\operatorname{PGL}(2,\mathbb{R}),D_k}$-space.

Rank $n$ swapping algebra for the $\operatorname{PSL}(n, \mathbb{R})$ Hitchin component