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Physics Letters A

Factorization of the ground-state wavefunctions of quantum systems related to semisimple lie algebras

V. I. Inozemtsev, D. V. Meshcheryakov

Published Apr 1, 1985• 1 citation
Publisher
DOI
10.1016/0375-9601(85)90103-3

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Lie algebras describe how control Hamiltonians combine in closed quantum systems. In open Markovian systems, however, dissipation introduces irreversible directions that the Lie algebra alone does not retain. A dynamical Lie wedge preserves this information as a convex cone of locally admissible generators. The classical Levi decomposition separates a finite-dimensional Lie algebra into semisimple and solvable parts. In this work, we prove a corresponding decomposition and reconstruction theorem for dynamical Lie wedges. The theorem decomposes the wedge into semisimple and solvable data, records how these components are coupled, and provides a converse reconstruction of the wedge. The framework yields four structural types of the generated algebra. We further prove that the total dissipation strength of every admissible generator depends only on its coordinate in the solvable radical. In particular, if the generated Lie algebra is semisimple, every admissible generator is Hamiltonian. Together, these results extend Lie-algebraic structural methods to irreversible Markovian control and clarify how dissipation is encoded in the generator geometry.
We present a method for associating labeled directed graphs to finite-dimensional Lie algebras that are graphgraphgraph-admissibleadmissibleadmissible. The proposed framework enables visual identification of key structural algebraic features. We first formalize this approach and establish several classes of Lie algebras that admit the categorization as graph-admissible Lie algebras, and then investigate suitable valid graphs for this approach by translating the antisymmetry of the Lie bracket and the Jacobi identity into rules that such graphs must satisfy. Furthermore, based on these foundations, we provide graph-theoretic criteria for solvability, nilpotency, presence of ideals, simplicity, semisimplicity, and reductiveness of an algebra. We also present practical algorithms for constructing such graphs and those associated with the lower central series and derived series via iterative pruning procedures. We further cover specific examples, such as the Schrödinger and Lorentz algebras, which illustrate the applicability of the developed tools to physically relevant cases. Moreover, we explore applications in physics, where the method facilitates computation of similarity transformations essential for determining quantum mechanical time evolution via the Lie-algebraic factorization method. We also discuss a possible generalization of the framework based on magma-gradings of Lie algebras and formulate related open questions. Our approach bridges algebraic and combinatorial perspectives, offering theoretical tools for this area of mathematical physics.
Every finite-dimensional real or complex Lie algebra has a largest solvable ideal, its radical. In the compact dynamical Lie algebra (DLA) of a closed quantum system, this radical is the center; projecting onto it isolates directions that commute with the supplied algebra. With sparse Lie-bracket data and a known spectral gap, we construct a quantum circuit that approximately encodes this coefficient-space projector. The general construction combines the derived-algebra map with the Killing form, whereas compactness makes the Killing form alone sufficient. This distinction matters for numerical sensitivity, and our main theorem gives its exact law under an invariant-orthonormal basis adapted to the center and semisimple part. In a compact real algebra with a nonzero semisimple part, the condition number of the general operator is the three-halves power of the Killing-form condition number on that part. The compact projector in turn yields a bounded-error test for whether the center is trivial, provided that any nonzero center has at least a stated minimum dimension. This gives a controlled test for internal conserved directions in compact quantum dynamics.