标题： 经典动力学$r$-矩阵在广义三维引力的Chern-Simons表述中的应用 显示英文标题

标题： Classical dynamical $r$-matrices for the Chern-Simons formulation of generalised 3d gravity

摘要： 经典的动力学$r$-矩阵自然出现在陈-西蒙斯理论相空间的组合描述中，这要么是通过包含动力学源，要么是通过涉及两个穿孔的规范固定程序实现的。 在这里，我们考虑陈-西蒙斯形式化三维引力中出现的一类李代数的经典动力学$r$-矩阵，适用于任何宇宙常数值。 在这种情况下，我们推导出经典动力学$r$-矩阵的微分方程，并表明它们可以被视为一种广义复化，按照我们定义的方式，这些方程控制着$\mathfrak{su}(2)$和$\mathfrak{sl}(2,\mathbb{R})$的动力学$r$-矩阵。 我们得到显式的解族，并通过魏尔斯特拉斯因子化方法将它们与 Feher、Gabor、Marshall、Palla 和 Pusztai 在手征 WZWN 模型背景下的解联系起来。 显示英文摘要

摘要： Classical dynamical $r$-matrices arise naturally in the combinatorial description of the phase space of Chern-Simons theories, either through the inclusion of dynamical sources or through a gauge-fixing procedure involving two punctures. Here we consider classical dynamical $r$-matrices for the family of Lie algebras which arise in the Chern-Simons formulation of 3d gravity, for any value of the cosmological constant. We derive differential equations for classical dynamical $r$-matrices in this case, and show that they can be viewed as generalised complexifications, in a sense which we define, of the equations governing dynamical $r$-matrices for $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{R})$. We obtain explicit families of solutions and relate them, via Weierstrass factorisation, to solutions found by Feher, Gabor, Marshall, Palla and Pusztai in the context of chiral WZWN models.