Brane Correspondence on Nilmanifolds

We formulate T-duality on nilmanifolds within generalized complex geometry and provide an explicit Lie algebraic description of the induced brane correspondence. Using invariant structures, the duality is realized as an isomorphism of Courant algebroids, enabling a concrete transport of generalized complex structures and brane data. We show that symplectic branes map to generalized complex branes in a precise computable way: dual symplectic branes appear as Lagrangian subalgebras, while coisotropic branes with flux undergo dimensional reduction, yielding explicit instances of brane transmutation. These results reveal a precise algebraic structure governing the interplay between flux, duality, and geometry. We further propose a conjectural characterization of brane subalgebras in type-changing backgrounds in terms of projections to the complex sector and associated coisotropic data.