We introduce a family of posets which generate Lie poset subalgebras of \(A_{n-1}=\mathfrak {sl}(n)\) whose index can be realized topologically. In particular, if \(\mathcal {P}\) is such a toral poset, then it has a simplicial realization which is homotopic to a wedge sum of d one-spheres, where d is the index of the corresponding type-A Lie poset algebra \(\mathfrak {g}_A(\mathcal {P})\). Moreover, when \(\mathfrak {g}_A(\mathcal {P})\) is Frobenius, its spectrum is binary, that is, consists of an equal number of 0’s and 1’s. We also find that all Frobenius, type-A Lie poset algebras corresponding to a poset whose largest totally ordered subset is of cardinality at most three have a binary spectrum.

我们介绍了一组姿势集，它们生成\（A_ {n-1} = \ mathfrak {sl}（n）\）的Lie姿势子代数，其索引可以通过拓扑实现。特别是，如果\（\ mathcal {P} \）是这样的环形姿势，则它具有与d个单球体的楔形和同义的简单实现，其中d是对应的A型李的索引位姿代数\（\ mathfrak {g} _A（\ mathcal {P}）\）。此外，当\（\ mathfrak {g} _A（\ mathcal {P}）\）为Frobenius时，其频谱为二进制即由相等的0和1组成。我们还发现，所有与最大总有序子集的基数最多为3的基态相对应的基态的所有Frobenius A型Lie代数代数都具有二元谱。