论文标题
通过$ n $ -lie代数共同学的杨百车操作员的变形
Deformations of Yang-Baxter operators via $n$-Lie algebra cohomology
论文作者
论文摘要
我们介绍了$ n $ - 元素自分配对象的张量类别中的矢量空间类别中的共同体学理论，该对象对其无限变形进行了分类。对于从$ n $ -lie代数获得的$ n $ -ary的自分配对象，我们表明（$ n $ - ary）Lie Coomomology自然会注入自分配的共同体学，我们在温和的假设下证明，该地图是第二个共同学组的同构。这表明自我分布变形完全由谎言支架的变形完全分类。该理论在研究Yang-Baxter操作员的研究中具有重要的应用，因为自分配变形确定了Yang-Baxter操作员的非平地变形，该Yang-baxter运算符源自$ n $ ary-ary-ary-ary-ary-ary-are-are ry-arter-arter-ary-ary-are-ary-are-ary-arter-arter-arter-are-are distrable offeration结构。特别是，我们表明，从第二个自我分布的共同体到相关的Yang-Baxter操作员的第二个共同体存在同态。此外，我们证明，当自分配结构由具有微不足道的中心的谎言代数诱导时，我们会得到单态性。我们基于同时变形构建了变形理论，在该变形的情况下，同时构造了山结构和自分配结构。我们表明，当Lie代数具有非平凡的共同体学（例如，对于半简单的LIKE代数）时，同时变形可能仍然是不繁琐的，会产生相应的Yang-Baxter操作员变形。我们在低维度中提供示例和计算，并且我们完全表征了从所有非平凡的真实谎言代数$ 3 $获得的$ 2 $ cocycles，即Bianchi I-ix，以及所有的非实用的尺寸尺寸的尺寸。
We introduce a cohomology theory of $n$-ary self-distributive objects in the tensor category of vector spaces that classifies their infinitesimal deformations. For $n$-ary self-distributive objects obtained from $n$-Lie algebras we show that ($n$-ary) Lie cohomology naturally injects in the self-distributive cohomology and we prove, under mild additional assumptions, that the map is an isomorphism of second cohomology groups. This shows that the self-distribuitve deformations are completely classified by the deformations of the Lie bracket. This theory has important applications in the study of Yang-Baxter operators as the self-distributive deformations determine nontrivial deformations of the Yang-Baxter operators derived from $n$-ary self-distributive structures. In particular, we show that there is a homomorphism from the second self-distributive cohomology to the second cohomology of the associated Yang-Baxter operator. Moreover, we prove that when the self-distributive structure is induced by a Lie algebra with trivial center, we get a monomorphism. We construct a deformation theory based on simultaneous deformations, where both the coalgebra and self-distributive structures are deformed simultaneously. We show that when the Lie algebra has nontrivial cohomology (e.g. for semi-simple Lie algebras) the simultaneous deformations might still be nontrivial, producing corresponding Yang-Baxter operator deformations. We provide examples and computations in low dimensions, and we completely characterize $2$-cocycles for the self-distributive objects obtained from all the nontrivial real Lie algebras of dimension $3$, i.e. the Bianchi I-IX, and all the nontrivial complex Lie algebras of dimension $3$.
