The Structure of the Ghost and Swapping Algebras

Abstract

The swapping algebras and the ghost algebras were introduced in the papers of Labourie and Bridgeman--Labourie to encode the Poisson structure on spaces of length functions and generalized cross-ratio functions associated to $Θ$-Anosov representations. They used these algebras to show that correlation functions are stable under the Poisson bracket associated to Atiyah-Bott-Goldman's symplectic form and that length functions are convex along the Hamiltonian flow of other length functions. In this paper, we study the algebraic relationship between these two algebras. In particular, we show that the kernel of a natural surjection from the ghost algebra onto the extended swapping algebra is generated by five families of relations. Consequently, we obtain an isomorphism from an explicit quotient of the ghost algebra onto the extended swapping algebra.