论文标题
模块化双重指数和重力模型中的应用
Quantum exponentials for the modular double and applications in gravity models
论文作者
论文摘要
在此注释中，我们提出了量子矩阵组SL $ _Q^+（2，\ Mathbb {r}）$的分解为（（变形）FADDEEV $ \ text {u} {u} _q（\ mathfrak sl} $ {$ mathbb的量子代数生成器的量子代数生成器的指示。通过将双曲线表示矩阵与Whittaker函数联系起来来检查公式。我们以Hopf二元性来解释（或推导）它，并使用它来明确构建模块化双重的定期表示，从而导致Casimir及其模块化双重二重性作为量子群上的Laplacian的模拟二重奏。此描述对于2D Liouville重力和3D纯重力都很重要，因为两者都受该代数结构的控制。该结果建立在这两种引力模型的幅度的$ q $ -bf公式中。
In this note, we propose a decomposition of the quantum matrix group SL$_q^+(2,\mathbb{R})$ as (deformed) exponentiation of the quantum algebra generators of Faddeev's modular double of $\text{U}_q(\mathfrak{sl}(2, \mathbb{R}))$. The formula is checked by relating hyperbolic representation matrices with the Whittaker function. We interpret (or derive) it in terms of Hopf duality, and use it to explicitly construct the regular representation of the modular double, leading to the Casimir and its modular dual as the analogue of the Laplacian on the quantum group manifold. This description is important for both 2d Liouville gravity, and 3d pure gravity, since both are governed by this algebraic structure. This result builds towards a $q$-BF formulation of the amplitudes of both of these gravitational models.
