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

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

2008-08-18

arxiv.org/abs/0808.2423v1

Mathematics

Quantum Algebra

Scientific paper

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.

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