2021/09/23 by Fahimeh Mokhtari, Mokhtari, Fahimeh, Jan A. Sanders +1
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Dynamical Systems (math.DS) #FOS: Mathematics #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2109.11419
openalex publication_date 2021/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 netC,N is isomorphic to the semidirect sum of a semisimple part, consisting of two simple components \mathfrakslC and \mathfrakslB, 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≃ glC and B in b ≃ glB, and an abelian algebra a≃\mathfrakGr(C,N), the Grassmannian, consisting of the C-dimensional subspaces of ℝN. The methods in this paper can be immediately applied to study the linear maps of colored networks.