English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this note, we propose a decomposition of the quantum matrix groupSL$_q^+(2,\mathbb{R})$ as (deformed) exponentiation of the quantum algebragenerators of Faddeev's modular double of $\text{U}_q(\mathfrak{sl}(2,\mathbb{R}))$. The formula is checked by relating hyperbolic representationmatrices with the Whittaker function. We interpret (or derive) it in terms ofHopf duality, and use it to explicitly construct the regular representation ofthe modular double, leading to the Casimir and its modular dual as the analogueof the Laplacian on the quantum group manifold. This description is importantfor both 2d Liouville gravity, and 3d pure gravity, since both are governed bythis algebraic structure. This result builds towards a $q$-BF formulation ofthe amplitudes of both of these gravitational models.

Quantum exponentials for the modular double and applications in gravity models