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 \textit{ 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 \textit{ 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.
Categories: math.RA