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The metric algebroid proposed by Vaisman (the Vaisman algebroid) governs thegauge symmetry algebra generated by the C-bracket in double field theory (DFT).We show that the Vaisman algebroid is obtained by an analogue of the Drinfel'ddouble of Lie algebroids. Based on a geometric realization of doubledspace-time as a para-Hermitian manifold, we examine exterior algebras and apara-Dolbeault cohomology on DFT and discuss the structure of the Drinfel'ddouble behind the DFT gauge symmetry. Similar to the Courant algebroid in thegeneralized geometry, Lagrangian subbundles $(L,\tilde{L})$ in a para-Hermitianmanifold play Dirac-like structures in the Vaisman algebroid. We find that analgebraic origin of the strong constraint in DFT is traced back to thecompatibility condition needed for $(L,\tilde{L})$ be a Lie bialgebroid. Theanalysis provides a foundation toward the "coquecigrue problem" for the gaugesymmetry in DFT.

Doubled Aspects of Vaisman Algebroid and Gauge Symmetry in Double Field Theory