  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 finite Morse
theory argument establishes that the simplicial realization of such posets is
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, the Lie poset algebra is
absolutely rigid and cannot be deformed. We also provide matrix representations
for Lie poset algebras in the other classical types. By so doing, we are able
to give examples of deformable Lie algebras which are both solvable and
Frobenius. This resolves a question of Gerstenhaber and Giaquinto about the
existence of such algebras.
