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The Lie algebraic structure of colored networks

Fahimeh Mokhtari, Jan Sanders

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.

The Lie algebraic structure of colored networks

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 -dimensional vector fields with C colors (different functions describing different types of cells in the network) this Lie algebra is isomorphic to the semidirect sum of a semisimple part, consisting of two simple components and , with , which we write as a block-matrix and a solvable part, consisting of two elements representing the identity in and B in , and an abelian algebra , the Grassmannian, consisting of the -dimensional subspaces of . The methods in this paper can be immediately applied to study the linear maps of colored networks.

Paper Structure

This paper contains 25 sections, 20 theorems, 136 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Normal form theory
  4. Semigroup(oid)
  5. Some motivating examples
  6. $N=3, {C}=1$
  7. $N=3=2\oplus 1$, ${C}=2$
  8. Structure theory in the colored case
  9. Subalgebra decomposition of $\mathfrak{net}_{{C},N}$
  10. The Lie algebra $\mathfrak{net}_{{C},N}$
  11. The Lie subalgebras $\mathfrak{{b}}^{}_{}$ and $\mathfrak{{c}}^{}_{}$
  12. The Lie subalgebra $\mathfrak{{a}}^{}_{}$
  13. The Lie subalgebra $\mathfrak{{o}}^{}_{}$
  14. The involution $\theta:\mathfrak{net}_{{C},N}\simeq\mathfrak{net}_{{B},N}$
  15. Structure of the Lie algebra $\mathfrak{net}_{{C},N}$
  16. ...and 10 more sections

Key Result

Lemma 4.6

The $\left\{\Sigma_k^l\right\}_{k,l=1}^{C}$ together form a semigroupoid $\Sigma$ and $\nu$ is a faithful antirepresentation of $\Sigma$ in $\mathfrak{gl}_N$.

Theorems & Definitions (66)

  • Remark 1.3
  • Example 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 2.1
  • Remark 2.2
  • Example 3.1
  • Remark 3.2
  • Example 3.3
  • ...and 56 more