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We investigate properties of a Type-A meander, here considered to be a certain planar graph associated to seaweed subalgebra of the special linear Lie algebra. Meanders are designed in such a way that the index of the seaweed may be computed by counting the number and type of connected components of the meander. Specifically, the simplicial homotopy types of Type-A meanders are determined in the cases where there exist linear greatest common divisor index formulas for the associate seaweed. For Type-A seaweeds, the homotopy type of the algebra, defined as the homotopy type of its associated meander, is recognized as a conjugation invariant which is more granular than the Lie algebra's index.

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We introduce a family of posets which generate Lie poset subalgebras of A n−1 = sl(n) whose index can be realized topologically. In particular, if 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 g A (P). Moreover, when g A (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. KeywordsFrobenius Lie algebra • Poset algebra • Spectrum • Index Mathematics Subject Classification 2010 17B99 • 05E15 ind g = min F∈g * dim(ker(B F )),

We define posets of types B, C, and D. These posets encode the matrix forms of certain Lie algebras which lie between the algebras of upper-triangular and diagonal matrices. Our primary concern is the index and spectral theories of such type-B, C, and D Lie poset algebras. For an important restricted class, we develop combinatorial index formulas and, in particular, characterize posets corresponding to Frobenius Lie algebras. In this latter case we show that the spectrum is binary; that is, consists of an equal number of 0's and 1's. Interestingly, type-B, C, and D Lie poset algebras can be related to Reiner's notion of a parset.