标题： 与SL$(2,\mathbb R)$和SL$(2,\mathbb R)/U(1)$相关的无限秩李代数 显示英文标题

标题： An infinite-rank Lie algebra associated to SL$(2,\mathbb R)$ and SL$(2,\mathbb R)/U(1)$

摘要： 我们通过从非紧流形${\cal M}=$SL$(2,\mathbb R)$和${\cal M}=$SL$(2,\mathbb R)/U(1)$到有限维单李群$G$的光滑映射，构建了一个广义的Kac-Moody代数概念。 该构造通过两种等价方式实现：通过Plancherel定理以及在$L^2(\mathcal{M})$中识别一个希尔伯特基。 我们分析了中心扩张的存在性，并确定了与$\cal M$上厄米算子对偶的那些扩张。 通过检查$\mathfrak{sl}(2,\mathbb{R})$的 Clebsch-Gordan 系数，我们推导出表征相应广义 Kac-Moody 代数的李括号。这些代数的根结构被确定，并且表明可以定义无限多个同时对易的算子。此外，我们简要提及这些代数在超引力领域中的应用，特别是在标量场坐标化非紧流形$\text{SL}(2,\mathbb{R})/U(1)$的情形中。 显示英文摘要

摘要： We construct a generalised notion of Kac-Moody algebras using smooth maps from the non-compact manifolds ${\cal M}=$SL$(2,\mathbb R)$ and ${\cal M}=$ SL$(2,\mathbb R)/U(1)$ to a finite-dimensional simple Lie group $G$. This construction is achieved through two equivalent ways: by means of the Plancherel Theorem and by identifying a Hilbert basis within $L^2(\mathcal{M})$. We analyse the existence of central extensions and identify those in duality with Hermitean operators on $\cal M$. By inspecting the Clebsch-Gordan coefficients of $\mathfrak{sl}(2,\mathbb{R})$, we derive the Lie brackets characterising the corresponding generalised Kac-Moody algebras. The root structure of these algebras is identified, and it is shown that an infinite number of simultaneously commuting operators can be defined. Furthermore, we briefly touch upon applications of these algebras within the realm of supergravity, particularly in scenarios where the scalar fields coordinatize the non-compact manifold $\text{SL}(2,\mathbb{R})/U(1)$.