增强的Leibniz代数：结构定理与诱导Lie 2-代数
Enhanced Leibniz Algebras: Structure Theorem and Induced Lie 2-Algebra
摘要：
增强莱布尼茨代数是一种代数结构，它出现在描述自相互作用格伯的特定高规范理论的背景下。它由一个Leibniz代数$(\mathbb{V},[ \cdot, \cdot ])$、一个在$\mathbb{V}$上取值于另一个向量空间$\mathbb{W}$的双线性型和一个映射$t \colon \mathbb{W} \to \mathbb{V}$组成，共满足四个相容关系.我们的结构定理证明了一个增强的Leibniz代数是由其基础Leibniz代数$(\mathbb{V},[ \cdot, \cdot ])$、它内部的一个适当的交换理想${\mathfrak i}$以及一个只影响$\mathbb{W}$ -值积的上同调2-类$[Δ]$唯一确定的.正二次增强的莱布尼茨代数，正如杨-米尔斯型作用泛函的定义所需要的那样，对基础的莱布尼茨代数$(\mathbb{V},[ \cdot, \dot ])$有相当大的限制：$\mathbb{V}$必须是正二次李代数${\mathfrak g}$与${\mathfrak g}$ -模${\mathfrak i}$的半半直积，$\mathbb{V} \cong {\mathfrak g}\ltimes{\mathfrak i}$，其中${\mathfrak i}$在这种情况下是上述理想。本文的第二个主要结果是构造一个函子，从这类增强的Leibniz代数的范畴到（半严格）Lie 2-代数的范畴，或者等价地，到两项$L_\infty$ -代数的范畴。
英文摘要：
An enhanced Leibniz algebra is an algebraic struture that arises in the context of particular higher gauge theories describing self-interacting gerbes. It consists of a Leibniz algebra $(\mathbb{V},[ \cdot, \cdot ])$, a bilinear form on $\mathbb{V}$ with values in another vector space $\mathbb{W}$, and a map $t \colon \mathbb{W} \to \mathbb{V}$, satisfying altogether four compatibility relations. Our structure theorem asserts that an enhanced Leibniz algebra is uniquely determined by the underlying Leibniz algebra $(\mathbb{V},[ \cdot, \cdot ])$, an appropriate abelian ideal ${\mathfrak i}$ inside it, as well as a cohomology 2-class $[Δ]$ which only effects the $\mathbb{W}$-valued product. Positive quadratic enhanced Leibniz algebras, as needed for the definition of a Yang-Mills type action functional, turn out to be rather restrictive on the underlying Leibniz algebra $(\mathbb{V},[ \cdot, \dot ])$: $\mathbb{V}$ has to be the hemisemidirect product of a positive quadratic Lie algebra ${\mathfrak g}$ with a ${\mathfrak g}$-module ${\mathfrak i}$, $\mathbb{V} \cong {\mathfrak g}\ltimes{\mathfrak i}$, with ${\mathfrak i}$ the above-mentioned ideal in this case. The second main result of this article is the construction of a functor from the category of such enhanced Leibniz algebras to the category of (semi-strict) Lie 2-algebras or, equivalentely, of two-term $L_\infty$-algebras.
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