The Lie algebraic structure of colored networks

TL;DR

This paper characterizes the Lie algebra structure of linear colored network vector fields, providing a concrete algorithm for Levi decomposition and revealing its isomorphism to a semidirect sum of simple, solvable, and abelian components.

Contribution

It introduces a concrete algorithm for the Levi decomposition of the Lie algebra of colored network vector fields and describes its explicit structure as a semidirect sum of known algebraic components.

Findings

The Lie algebra of colored network vector fields is isomorphic to a semidirect sum of simple, solvable, and abelian parts.

The algorithm can be applied to study linear maps of colored networks.

The algebraic structure includes components like sl}_C, sl}_B, and the Grassmannian.

Abstract

In the computation of the normal form of a colored network vector field, following the semigroup(oid) approach in [19], one would like to be able to say something about the structure of the Lie algebra of linear colored network vector fields. Unlike the purely abstract approach in [10], we describe here a concrete algorithm that gives us the Levi decomposition. If we apply this algorithm to a given subalgebra, it does put the elements in the subalgebra in the block form given by the Levi decomposition, but this need not be the Levi decomposition of the given subalgebra. We show that for $N$-dimensional vector fields with C colors (different functions describing different types of cells in the network) this Lie algebra $net_{C,N}$ is isomorphic to the semidirect sum of a semisimple part, consisting of two simple components $\mathfrak{sl}_C$ and $\mathfrak{sl}_B$, with $B=N-C$, which we write as a block-matrix and a solvable part, consisting of two elements representing the identity $C$ in $c\simeq glC$ and B in $b \simeq glB$, and an abelian algebra $a\simeq\mathfrak{Gr}(C,N)$, the Grassmannian, consisting of the $C$-dimensional subspaces of $\mathbb{R}^N$. The methods in this paper can be immediately applied to study the linear maps of colored networks.