AI 中文总结
本研究针对马尔可夫量子动力学的动力学李楔，证明了类莱维分解与重构定理，揭示了耗散在生成元几何中的编码方式，拓展了李代数结构方法至不可逆马尔可夫控制领域。
AI 中文摘要
李代数描述了控制哈密顿量在封闭量子系统中的组合方式，然而在开放马尔可夫系统中，耗散引入了李代数无法单独保留的不可逆方向。动力学李楔作为局部可允许生成元的凸锥，保留了这一信息。经典莱维分解将有限维李代数拆分为半单部分与可解部分，本研究针对动力学李楔证明了对应的分解与重构定理：该定理将楔分解为半单与可解数据，记录各分量的耦合方式，并提供楔的逆重构，该框架可生成所生成代数的四种结构类型。我们进一步证明，每个可允许生成元的总耗散强度仅取决于其在可解根基中的坐标；特别地，若所生成的李代数为半单，则所有可允许生成元均为哈密顿量。这些结果将李代数结构方法拓展至不可逆马尔可夫控制领域，阐明了耗散如何编码于生成元的几何结构中。
英文摘要
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.
