Rank n swapping algebra for the PSL(n;R) Hitchin component

Zhe Sun Yau Mathematical Sciences Center

Combinatorics Differential Geometry mathscidoc:1609.06001

International Mathematics Research Notices, 2017, (2), 583-613, 2017.2
F. Labourie [arXiv:1212.5015] characterized the Hitchin components for PSL(n,R) for any n>1 by using the swapping algebra, where the swapping algebra should be understood as a ring equipped with a Poisson bracket. We introduce the 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 the PSL(n,R) Hitchin component for a fixed n>1. We also show the relation between the rank 2 swapping algebra and the cluster X space.
Hitchin component, rank n, swapping algebra, Fock-Goncharov, cluster.
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  title={Rank n swapping algebra for the PSL(n;R) Hitchin component},
  author={Zhe Sun},
  booktitle={International Mathematics Research Notices},
Zhe Sun. Rank n swapping algebra for the PSL(n;R) Hitchin component. 2017. Vol. 2017. In International Mathematics Research Notices. pp.583-613. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160902154652944666611.
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