Symplectic

Lie groupoids and Lie algebroids arise as generalizations of Lie groups and Lie algebras, extending the interplay between global and infinitesimal objects to a broader setting. In this context, new geometric structures emerge: when considering Drinfeld doubles of Lie bialgebroids, unlike the case of Lie algebras, where the Drinfeld double is again a Lie algebra, the algebroid counterpart carries a different type of structure, namely a Courant algebroid. Motivated by Poisson Lie theory, one is naturally led to consider double structures, where two geometric objects such as Lie groupoids, Lie algebroids, or Courant algebroids are combined in a compatible manner. In this talk, we discuss these double structures and how Manin triple descriptions are extended to this setting, highlighting recent results.