The metric algebroid proposed by Vaisman (the Vaisman algebroid) governs the gauge symmetry algebra generated by the C-bracket in double field theory (DFT). We show that the Vaisman algebroid is obtained by an analog of the Drinfel’d double of Lie algebroids. Based on a geometric realization of doubled space-time as a para-Hermitian manifold, we examine exterior algebras and a para-Dolbeault cohomology on DFT and discuss the structure of the Drinfel’d double behind the DFT gauge symmetry. Similar to the Courant algebroid in the generalized geometry, Lagrangian sub-bundles in a para-Hermitian manifold play Dirac-like structures in the Vaisman algebroid. We find that an algebraic origin of the strong constraint in DFT is traced back to the compatibility condition needed for to be a Lie bialgebroid. The analysis provides a foundation toward the “coquecigrue problem” for the gauge symmetry in DFT.
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