论文标题
从谎言代数交叉模块到张量层次结构
From Lie algebra crossed modules to tensor hierarchies
论文作者
论文摘要
本文虽然灵感来自于在理论物理学中使用张量层次结构的启发，但仍建立了其数学凭证，尤其是与lie代数交叉模块相关的基因相关。超级重力中的测量程序依赖于莱布尼兹代数和谎言代数之间的配对 - 嵌入张量。两个这样的代数及其嵌入张量形成了三重，称为lie-leibniz三重，其中包含代数的模块是特定的情况。本文致力于表明，任何lie -leibniz三重都诱导了一个差异分级的谎言代数（其相关的张量层次结构），其限制限制了Lie代数交叉模块类别的类别，这是与任何lie代数交叉模块相关的规范分配，其与任何相应的独特的2 -Term差异级别的lie lie lie liegerbra相关联。这表明，Lie-Leibniz三元组形成了谎言代数的天然概括，并且它们相关的张量层次结构可以被视为前者的某种“谎言化”。我们认为，与以前的推导相比，目前的张量层次结构的结构更清晰，更直接。我们强调，这种结构表明存在进一步定义明确的莱布尼兹仪理论。
The present paper, though inspired by the use of tensor hierarchies in theoretical physics, establishes their mathematical credentials, especially as genetically related to Lie algebra crossed modules. Gauging procedures in supergravity rely on a pairing - the embedding tensor - between a Leibniz algebra and a Lie algebra. Two such algebras, together with their embedding tensor, form a triple called a Lie-Leibniz triple, of which Lie algebra crossed modules are particular cases. This paper is devoted to showing that any Lie-Leibniz triple induces a differential graded Lie algebra - its associated tensor hierarchy - whose restriction to the category of Lie algebra crossed modules is the canonical assignment associating to any Lie algebra crossed module its corresponding unique 2-term differential graded Lie algebra. This shows that Lie-Leibniz triples form natural generalizations of Lie algebra crossed modules and that their associated tensor hierarchies can be considered as some kind of 'lie-ization' of the former. We deem the present construction of such tensor hierarchies clearer and more straightforward than previous derivations. We stress that such a construction suggests the existence of further well-defined Leibniz gauge theories.
