Quasi-Poisson varieties from double quasi-Poisson algebras in types $B,C,D$

Abstract: Double (quasi-)Poisson brackets were introduced on associative algebras by Van den Bergh to induce a (quasi-)Poisson structure on their representation spaces naturally equipped with a $\mathrm{GL}$-action (type $\mathtt{A}$). If there exists a compatible involutive anti-automorphism on the underlying associative algebras, Olshanski and Safonkin proved that this construction can be upgraded to induce a Poisson structure on twisted representation spaces (types $\mathtt{B},\mathtt{C},\mathtt{D}$). We provide an analogous result for double quasi-Poisson brackets, and over an arbitrary semisimple base. We also apply our theory to quivers in order to understand the Poisson structure on twisted (localised multiplicative) quiver varieties. The formalism permits that different vertices are assigned different types. As a first application, we recover the framework of Massuyeau and Turaev for Hopf algebras with a Fox pairing, which induces in particular the Poisson structure of character varieties for the orthogonal or symplectic groups. As a second application, we introduce a modified Kontsevich system.