More on doubled aspects of algebroids in double field theory

Journal of Mathematical Physics, Volume 61, Issue 12, December 2020. We continue to study doubled aspects of algebroid structures equipped with the C-bracket in double field theory (DFT). We find that a family of algebroids, the Vaisman (metric or pre-DFT), the pre- and the ante-Courant algebroids are constructed by the analogue of the Drinfel’d double of Lie algebroid pairs. We examine geometric implementations of these algebroids in the para-Hermitian manifold, which is a realization of the doubled space–time in DFT. We show that the strong constraint in DFT is necessary to realize the doubled and non-trivial Poisson structures but can be relaxed for some algebroids. The doubled structures of twisted brackets and those associated with group manifolds are briefly discussed.