| 摘要: |
| 级联衰变链方程也叫贝特曼方程，本文从表示论的角度研究~$n$~能级级联衰变链方程的解空间结构。
通过将系数矩阵视为下三角矩阵代数~($\mathbf{A}_n$ 型遗传代数)~中的元素，
并利用模值函数空间的张量分解~$\mathbf{Fun}(S,V)\cong\mathbf{Fun}(S,\mathbb{F})\otimes_{\mathbb{F}}V$，
将衰变链方程等价地改写为标准同态方程~$D_V f-(I_n\otimes M)\cdot f=0$，
本文证明了无论衰变常数是常数还是时变函数，级联衰变链方程的齐次解空间的维数都恒等于~$n$，
并且该齐次解空间具有左模结构。 |
| 关键词: 级联衰变链 贝特曼方程 下三角矩阵代数 模值同态方程 解空间模结构 |
| DOI： |
| 分类号:O15 |
| 基金项目:国家自然科学基金项目 |
|
| Module structure of the solution space of Cascade Decay Chain Equations over algebras of type A |
|
Mengyun Wu, Yu-Zhe Liu
|
| Abstract: |
| The cascade decay chain equations are also known as the Bateman equations. In this paper, we study the structure of the solution space of the $n$-level cascade decay chain equations from the perspective of representation theory. By regarding the coefficient matrix as an element of the lower triangular matrix algebra (a hereditary algebra of type $\mathbf{A}_n$) and using the tensor decomposition $\mathbf{Fun}(S,V)\cong\mathbf{Fun}(S,\mathbb{F})\otimes_{\mathbb{F}}V$ of module-valued function spaces, we rewrite the decay chain equations equivalently as the standard homomorphic equation $D_V f-(I_n\otimes M)\cdot f=0$. We prove that, regardless of whether the decay constants are constant or time-dependent, the dimension of the homogeneous solution space of the cascade decay chain equations is always $n$, and moreover this homogeneous solution space carries a left module structure. |
| Key words: Cascade decay chain Bateman equations lower triangular matrix algebra module-valued homomorphic equations module structure of solution space |
