2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/94630We define a Fourier-Mukai transform for a triple consisting of two holomorphic vector bundles over an elliptic curve and a homomorphism between them. We prove that in some cases the transform preserves the natural stability condition for a triple. We also define a Nahm transform for solutions to natural gauge-theoretic equations on a triple -- vortices -- and explore some of its basic properties. Our approach combines direct methods with dimensional reduction techniques, relating triples over a curve with vector bundles over the product of the curve with the complex projective line.39 pages, LaTeX2e, no figures; new proofs added, some arguments rewritten and typos corrected. Final version to appear in Journal of Geometry and PhysicsAlgebraic GeometryDifferential Geometry14D20; 14H60; 14J60; 14H21text