2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/142236The moduli space of $G$-bundles on an elliptic curve with additional flag structure admits a Poisson structure. The bivector can be defined using double loop group, loop group and sheaf cohomology constructions. We investigate the links between these methods and for the case $\SL_2$ perform explicit computations, describing the bracket and its leaves in detail.Final version: typos fixed, some re-organization; to appear IJMAlgebraic GeometrySymplectic Geometry14H60, 53D17, 53D20text