Graphs, Frobenius functionals, and the classical Yang-Baxter equation

TL;DR

This paper introduces a cyclic functional on certain parabolic subalgebras of sl(n), demonstrating its non-degeneracy via graph properties, and explores solutions to the classical Yang-Baxter equation linked to these graphs.

Contribution

It defines a new cyclic functional on P(n,m), proves its non-degeneracy using graph theory, and connects these graphs to solutions of the classical Yang-Baxter equation.

Findings

Cyclic functional F is non-degenerate for P(n,m) when (n,m)=1

Graphs associated with F enable explicit solutions to the classical Yang-Baxter equation

Reconstruction of graphs from their local rings is possible

Abstract

A Lie algebra is Frobenius if it admits a linear functional F such that the Kirillov form F([x,y]) is non-degenerate. If g is the m-th maximal parabolic subalgebra P(n,m) of sl(n) this occurs precisely when (n,m) = 1. We define a "cyclic" functional F on P(n,m) and prove it is non-degenerate using properties of certain graphs associated to F. These graphs also provide in some cases readily computable associated solutions of the classical Yang-Baxter equation. We also define a local ring associated to each connected loopless graph from which we show that the graph can be reconstructed. Finally, we examine the seaweed Lie algebras of Dergachev and Kirillov from our perspective.