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.

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更多关于双场论中代数的双重方面

我们继续研究在双场理论 (DFT) 中配备 C 括号的代数体结构的双重方面。我们发现一个代数族，Vaisman（度量或前 DFT），前和前 Courant 代数是由李代数对的 Drinfel'd 对偶的类似物构建的。我们研究了这些代数在准厄米流形中的几何实现，这是 DFT 中双倍时空的实现。我们表明，DFT 中的强约束是实现双重和非平凡泊松结构所必需的，但对于某些代数可以放宽。简要讨论了扭曲支架的双重结构和与组流形相关的那些。