No abstract

We extend the set of known infinite families of Frobenius seaweed Lie subalgebras of sln to include a family which is the first non-trivial general family containing algebras whose associated meanders have an arbitrarily large number of parts.

Analogous to the sl(n) case, we address the computation of the index of seaweed subalgebras of sp(2n) by introducing graphical representations called symplectic meanders. Formulas for the algebra's index may be computed by counting the connected components of its associated meander. In certain cases, formulas for the index can be given in terms of elementary functions. Mathematics Subject Classification 2010 : 17B08, 17B20The results of this paper were delivered by the second author in a talk entitled Symplectic Meanders in a January 2015 conference at the University of Miami in honor of his adviser Michelle Wachs. Some of these results, inclusive of the meander construction, have been independently obtained by D. Panyushev and O. Yakimova per a recent arXiv post on January 3, 2016 [12].

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.

If g is a Frobenius Lie algebra, then for certain F ∈ g * the natural map g −→ g * given by x −→ F [x, −] is an isomorphism. The inverse image of F under this isomorphism is called a principal element. We show that if g is a Frobenius seaweed subalgebra of An−1 = sl(n) then the spectrum of the adjoint of a principal element consists of an unbroken set of integers whose multiplicites have a symmetric distribution. Our proof methods are constructive and combinatorial in nature.Mathematics Subject Classification 2010 : 17B20, 05E15