论文标题
多项式海森贝格代数的仿射Weyl组表征
An affine Weyl group characterization of polynomial Heisenberg algebras
论文作者
论文摘要
我们研究称为多项式海森贝格代数（PHAS）的谐波振荡器代数的变形，并在它们之间建立连接与$ a^{（1）} _ m $的延长仿型Weyl组，其中$ m $是PHA的程度。为了建立这种联系，我们采用超对称量子力学首先将多项式海森贝格代数连接到微分方程的对称系统。此连接以前已用于将量子系统与非线性微分方程相关联；最值得注意的是，第四和第五painlevé方程。完成此操作后，我们将使用有关Painlevé方程的Bäcklund变换的先前研究，其对称形式的概括为以扩展的仿生韦伊尔基团为特征。这项工作有助于更好地理解量子系统和代数结构来表征它们。
We study deformations of the harmonic oscillator algebra known as polynomial Heisenberg algebras (PHAs), and establish a connection between them and extended affine Weyl groups of type $A^{(1)}_m$, where $m$ is the degree of the PHA. To establish this connection, we employ supersymmetric quantum mechanics to first connect a polynomial Heisenberg algebra to symmetric systems of differential equations. This connection has been previously used to relate quantum systems to non-linear differential equations; most notably, the fourth and fifth Painlevé equations. Once this is done, we use previous studies on the Bäcklund transformations of Painlevé equations and generalizations of their symmetric forms characterized by extended affine Weyl groups. This work contributes to better understand quantum systems and the algebraic structures characterizing them.
