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@ARTICLE{Heib:1052370,
      author       = {Heib, Tim and Bruschi, David Edward},
      title        = {{O}n the structural properties of {L}ie algebras via
                      associated labeled directed graphs},
      publisher    = {arXiv},
      reportid     = {FZJ-2026-00964, 2601.16161},
      year         = {2026},
      abstract     = {We present a method for associating labeled directed graphs
                      to finite-dimensional Lie algebras, thereby enabling rapid
                      identification of key structural algebraic features. To
                      formalize this approach, we introduce the concept of
                      graph-admissible Lie algebras and analyze properties of
                      valid graphs given the antisymmetry property of the Lie
                      bracket as well as the Jacobi identity. Based on these
                      foundations, we develop graph-theoretic criteria for
                      solvability, nilpotency, presence of ideals, simplicity,
                      semisimplicity, and reductiveness of an algebra. Practical
                      algorithms are provided for constructing such graphs and
                      those associated with the lower central series and derived
                      series via an iterative pruning procedure. This visual
                      framework allows for an intuitive understanding of Lie
                      algebraic structures that goes beyond purely visual
                      advantages, since it enables a simpler and swifter grasping
                      of the algebras of interest beyond computational-heavy
                      approaches. Examples, which include the Schrödinger and
                      Lorentz algebra, illustrate the applicability of these tools
                      to physically relevant cases. We further explore
                      applications in physics, where the method facilitates
                      computation of similtude relations essential for determining
                      quantum mechanical time evolution via the Lie algebraic
                      factorization method. Extensions to graded Lie algebras and
                      related conjectures are discussed. Our approach bridges
                      algebraic and combinatorial perspectives, offering both
                      theoretical insights and computational tools into this area
                      of mathematical physics.},
      keywords     = {Mathematical Physics (math-ph) (Other) / Quantum Physics
                      (quant-ph) (Other) / FOS: Physical sciences (Other)},
      cin          = {PGI-12},
      cid          = {I:(DE-Juel1)PGI-12-20200716},
      pnm          = {5221 - Advanced Solid-State Qubits and Qubit Systems
                      (POF4-522) / Verbundprojekt: German Quantum Computer based
                      on Superconducting Qubits (GEQCOS) - Teilvorhaben:
                      Charakterisierung, Kontrolle und Auslese (13N15685)},
      pid          = {G:(DE-HGF)POF4-5221 / G:(BMBF)13N15685},
      typ          = {PUB:(DE-HGF)25},
      doi          = {10.48550/ARXIV.2601.16161},
      url          = {https://juser.fz-juelich.de/record/1052370},
}