AI 中文总结
研究给定狄拉克结构的狄拉克补问题，证明拉格朗日或局部狄拉克补无障碍并举例说明其复杂性，引入上同调类判断狄拉克补存在性，还用李理论技术证明特定李代数情形下对角线不存在补。
AI 中文摘要
给定狄拉克结构的狄拉克补的存在性是库朗代数胚结构理论和狄拉克结构变形理论中的核心问题。我们详细研究此问题，证明拉格朗日或局部狄拉克补无障碍，并给出示例展示该问题的复杂性。引入一个上同调类，其非零阻止狄拉克补的存在并应用于多个示例族。另一方面，利用李理论技术证明，对于赋予正定形式的李代数\(\mathfrak{g}\)，\(\mathfrak{g} \oplus \bar{\mathfrak{g}}\)中的对角线不存在补，除非\(\mathfrak{g}\)是阿贝尔的，这包括具有基灵形式的实紧半单李代数。
英文摘要
The existence of a Dirac complement for a given Dirac structure is a central question in the structure theory of Courant algebroids and the deformation theory of Dirac structures. We study this problem in detail, proving the unobstructedness of lagrangian or local Dirac complements and providing examples that show the complexity of this question. We introduce a cohomology class whose nonvanishing prevents the existence of a Dirac complement and apply it to several families of examples. On the other hand, by using Lie-theoretical techniques, we prove that, for a Lie algebra $\mathfrak{g}$ endowed with a definite form, the diagonal in $\mathfrak{g} \oplus \bar{\mathfrak{g}}$ does not admit a complement unless $\mathfrak{g}$ is abelian. This includes real compact semisimple Lie algebras with their Killing form.
Comments30 pages
