论文标题
Lie-Poisson代数和手性的变形
Deformation of Lie-Poisson algebras and chirality
论文作者
论文摘要
Linearization of a Hamiltonian system around an equilibrium point yields a set of Hamiltonian-symmetric spectra: If $λ$ is an eigenvalue of the linearized generator, $-λ$ and $\barλ$ (hence, $-\barλ$) are also eigenvalues -- the former implies a time-reversal symmetry, while the latter guarantees the reality of the solution.然而，围绕单数平衡点（通常存在于非规范性的哈密顿系统中）的线性化作用有所不同，从而破坏了光谱的哈密顿对称性。时间反转不对称会导致手性。这种有趣的现象首先是在分析Rattleback的手性运动时发现的，这是船形的顶部，惯性和几何形状的轴未对准[Phys。 Lett。 A 381（2017），2772--2777]。为了阐明手性光谱的产生方式，我们研究了三维lie-poisson系统，并对引起对称性破坏的奇异性原型进行了分类。核心思想是基础谎言代数的变形；我们援引比安奇（Bianchi）的所有三维LIE代数列表，我们表明，所谓的B类代数是由简单代数SO（3）的不对称变形产生的，当围绕其奇异性线性线性时，会产生手性手性光谱。变形理论被推广到更高的维度，包括与流体力学相关的无限二维泊松歧管。
Linearization of a Hamiltonian system around an equilibrium point yields a set of Hamiltonian-symmetric spectra: If $λ$ is an eigenvalue of the linearized generator, $-λ$ and $\barλ$ (hence, $-\barλ$) are also eigenvalues -- the former implies a time-reversal symmetry, while the latter guarantees the reality of the solution. However, linearization around a singular equilibrium point (which commonly exists in noncanonical Hamiltonian systems) works out differently, resulting in breaking of the Hamiltonian symmetry of spectra; time-reversal asymmetry causes chirality. This interesting phenomenon was first found in analyzing the chiral motion of the rattleback, a boat-shaped top having misaligned axes of inertia and geometry [Phys. Lett. A 381 (2017), 2772--2777]. To elucidate how chiral spectra are generated, we study the 3-dimensional Lie-Poisson systems, and classify the prototypes of singularities that cause symmetry breaking. The central idea is the deformation of the underlying Lie algebra; invoking Bianchi's list of all 3-dimensional Lie algebras, we show that the so-called class-B algebras, which are produced by asymmetric deformations of the simple algebra so(3), yield chiral spectra when linearized around their singularities. The theory of deformation is generalized to higher dimensions, including the infinite-dimensional Poisson manifolds relevant to fluid mechanics.
