We provide general closed-form formulas for the index of type-A Lie poset algebras corresponding to posets of restricted height. Furthermore, we provide a combinatorial recipe for constructing all posets corresponding to type-A Frobenius Lie poset algebras of heights zero, one, and two. A discrete Morse theory argument establishes that the simplicial realizations of such posets are contractible. It then follows, from a recent theorem of Coll and Gerstenhaber, that the second Lie cohomology group of the corresponding Lie poset algebra with coefficients in itself is zero. Consequently, such a Lie poset algebra is absolutely rigid and cannot be deformed. We also provide matrix representations for Lie poset algebras in the other classical types.

我们提供了与受限高度的球型相对应的A型李式球型代数索引的一般封闭式公式。此外，我们提供了一个组合配方，用于构造与高度为0、1和2的A型Frobenius Lie姿势代数相对应的所有姿势。离散的摩尔斯理论论证确立了这种摆姿的简单实现是可收缩的。然后，从Coll和Gerstenhaber的一个最新定理可以得出，相应的本身具有系数的Lieposet代数的第二个Lie同调群为零。因此，这种李·波塞特代数绝对是刚性的并且不能变形。我们还提供其他经典类型的Lieposet代数的矩阵表示。