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Labourie [ 19 ] characterized the Hitchin components forPSL  (  n  ,  ℝ  )for anyn  &gt;  1by using swapping algebra, where the swapping algebra is understood as a ring equipped with a Poisson bracket. We introduce rank  n  swapping algebra, which is the quotient of the swapping algebra by the(  n  +  1  )  ×  (  n  +  1  )determinant relations. The main results are the well-definedness of the rank  n  swapping algebra and the “cross-ratio” in its fraction algebra. As a consequence, we use the sub-fraction algebra of the rank  n  swapping algebra generated by these “cross-ratios” to characterize thePSL  (  n  ,  ℝ  )Hitchin component for a fixedn  &gt;  1    . We also show the relation between the rank 2 swapping algebra and the clusterX    PGL  (  2  ,  ℝ  )  ,    D   k-space.

Rank n Swapping Algebra for the PSL( n , ℝ ) Hitchin Component