论宇宙台球的隐藏超对称性
On a Hidden Supersymmetry of Cosmological Billiards
AI总结：
本文研究双曲李代数与几乎仿射李超代数的对应关系，将部分双曲李代数超对称化，列出97对可超对称化代数，分类不可对称化嘉当矩阵双曲李代数的超对称化及相关李超代数对应关系。
摘要：
在Belinski-Khalatnikov-Lifshitz宇宙学模型中，针对奇点采用振荡近似，宇宙台球的轨迹（无论其具体形式）在双曲李代数的外尔腔中实现。这类双曲李代数由Li Wang Lai完成分类；其超对称类似物——几乎仿射李超代数则在arXiv:0906.1860中完成分类。本文中，我们发现142个双曲李代数H中的部分可“超对称化”为几乎仿射李超代数S(H)：对每个此类H，可将其嘉当矩阵的某一行除以2，同时改变对应的谢瓦莱生成元的奇偶性及其间的关系。对于具有可对称化嘉当矩阵的H，我们列出全部97对这样的对应关系（H⟷S(H)）。在总共66个可“超对称化”的代数H中，有18个具有多个此类“超对称化”形式。对应于这些对的两个项（H⟷S(H)）的宇宙台球轨迹重合，因为H和S(H)拥有完全相同的外尔腔。我们还对具有不可对称化嘉当矩阵的双曲李代数的“超对称化”，以及简单有限维李超代数g（无嘉当矩阵且单分量为g_{\bar0}）的对应关系g⟷g_{\bar0}进行了分类。
英文摘要：
In the Belinski-Khalatnikov-Lifshitz cosmological models with an oscillatory approximation to singularity, the tracks of cosmological billiards (whatever they are) are realized in the Weyl chambers of the hyperbolic Lie algebras. The latter were classified by Li Wang Lai; their super analogs - the almost affine Lie superalgebras - were classified in arXiv:0906.1860. Here, we observe that some of the 142 hyperbolic Lie algebras $H$ can be ''superized'' to almost affine Lie superalgebras $S(H)$: for each of these $H$, it is possible to divide a row of its Cartan matrix by 2, simultaneously changing the parity of the corresponding Chevalley generators and relations between them. For the $H$ with a symmetrizable Cartan matrix, we list all 97 such pairs ($H\longleftrightarrow S(H)$). Several (18 of the total 66) thus ''superizable'' algebras $H$ have multiple such ''superizations''. The tracks of cosmological billiards corresponding to both terms of these pairs ($H\longleftrightarrow S(H)$) coincide since $H$ and $S(H)$ have one and the same Weyl chamber. We also classify ''superizations'' of hyperbolic Lie algebras with non-symmetrizable Cartan matrix, and pairs $\mathfrak{g}\longleftrightarrow \mathfrak{g}_{\bar 0}$, where $\mathfrak{g}$ is a simple finite-dimensional Lie superalgebra without Cartan matrix and simple component $\mathfrak{g}_{\bar 0}$.
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