Meanders and Frobenius Seaweed Lie Algebras

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.

Symplectic meanders

“…, a k ) | (n)), where k ≥ 4. This establishes that the formulas in [3] are, in some sense, the only "nice" ones.…”

“…The remaining move (component elimination) changes the index of the meander by eliminating a set of connected components. See [3] for details and examples. It follows from Theorem 2.1 that a seaweed is Frobenius precisely when its associated meander is homotopically trivial.…”

“…This led to the signature which we observe is, in essence, a graph theoretic rendering of Panyushev's reduction. In [3], and using the signature, Coll et al established the following extension of Elashvili's theorem.…”

Meander Graphs and Frobenius Seaweed Lie Algebras III

“…The sequence of moves used in this winding-down procedure is called the signature of the meander and may be regarding as a graph theoretic rendering of Panyushev's well-known reduction [6]. In [7,8], Coll et al used the signature to develop closed form formulas for the index of a seaweed algebra in terms of the block sizes of the defining flags in the Type-A and Type-C cases when the number of blocks in the flags is small. Subsequently, Karnauhova and Liebsher [10] used signature type moves and complexity arguments to establish that the index formulas developed in these papers are the only linear greatest common divisor formulas for the index based on the flags defining the seaweed.…”

“…Subsequently, Karnauhova and Liebsher [10] used signature type moves and complexity arguments to establish that the index formulas developed in these papers are the only linear greatest common divisor formulas for the index based on the flags defining the seaweed. One finds that in the Type-A case, a seaweed is Frobenius precisely when its associated meander consists of a single path [4,7,8]. For a Type-C seaweed to be Frobenius, its associated meander must reduce to a certain collection of paths [5,10].…”