We define a supersymmetric quantum mechanics of fermions that take values in a simple Lie algebra ##IMG## [http://ej.iop.org/images/1751-8121/53/42/425401/aabb0c1ieqn3.gif] {$\mathfrak{g}$} . We summarize what is known about the spectrum and eigenspaces of the Laplacian which corresponds to the Koszul differential d . Firstly, we concentrate on the zero eigenvalue eigenspace which coincides with the Lie algebra cohomology. We provide physical insight into useful tools to compute the cohomology, namely Morse theory and the Hochschild–Serre spectral sequence. We list explicit generators for the Lie algebra cohomology ring. Secondly, we concentrate on the eigenspaces of the supersymmetric quantum mechanics with maximal eigenvalue at given fermion number. These eigenspaces have an explicit description in terms of abelian ideals of a Borel subalgebra of the simple Lie algebra. We also introduce a model of Lie algebra valued fermions in two dimensions, where...

中文翻译：

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李代数费米子

我们定义了费米子的超对称量子力学，该费米子在简单的李代数中获得值## IMG ## [http://ej.iop.org/images/1751-8121/53/42/425401/aabb0c1ieqn3.gif] {$ \ mathfrak {g} $}。我们总结了对应于Koszul微分d的Laplacian谱和本征空间的已知信息。首先，我们集中于零特征值特征空间，该特征空间与李代数同调一致。我们提供了物理上的洞察力，以计算有用的工具来计算同调，即摩尔斯理论和Hochschild-Serre光谱序列。我们列出了李代数同调环的显式生成器。其次，我们关注在给定的费米子数下具有最大特征值的超对称量子力学的本征空间。这些本征空间具有关于简单李代数的Borel子代数的阿贝尔理想的明确描述。我们还介绍了二维的李代数值费米子模型，其中...