Rank n swapping algebra for PGLn Fock--Goncharov X moduli space

English

Sun, Zhe[University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]

Feb-2019

No

[en] Poisson algebra homomorphism ; rank n swapping algebra ; Fock–Goncharov X moduli space

[en] The rank n swapping algebra is a Poisson algebra defined on the set of ordered pairs of points of the circle using linking numbers, whose geometric model is given by a certain subspace of (K^n×K^{n∗})^r/GL(n,K). For any ideal triangulation of D_k---a disk with k points on its boundary, using determinants, we find an injective Poisson algebra homomorphism from the fraction algebra generated by the Fock--Goncharov coordinates for X_{PGL_n,D_k} to the rank n swapping multifraction algebra for r=k⋅(n−1) with respect to the (Atiyah--Bott--)Goldman Poisson bracket and the swapping bracket. This is the building block of the general surface case. Two such injective Poisson algebra homomorphisms related to two ideal triangulations T and T′ are compatible with each other under the flips.