Lie bialgebra structures on 2-step nilpotent graph algebras

TL;DR

This paper characterizes Lie bialgebra structures on 2-step nilpotent graph algebras, extending known results on Heisenberg algebras, and provides combinatorial criteria for these structures with applications to free 2-step nilpotent Lie algebras.

Contribution

It introduces a combinatorial approach to classify Lie bialgebra cobrackets on 2-step nilpotent graph algebras, generalizing previous results on Heisenberg algebras.

Findings

Restrictions on Lie bialgebra cobrackets for 2-step nilpotent algebras.

Graph-based combinatorial criteria for these structures.

Applications to free 2-step nilpotent Lie algebras.

Abstract

We generalize a result on the Heisenberg Lie algebra that gives restrictions to possible Lie bialgebra cobrackets on 2-step nilpotent algebras with some additional properties. For the class of 2-step nilpotent Lie algebras coming from graphs, we describe these extra properties in a very easy graph-combinatorial way. We exhibit applications for $\mathfrak f_n$, the free 2-step nilpotent Lie algebra.