Three-Dimensional Quantum Gravity from the Quantum Pseudo-Kähler Plane

Hyun Kyu Kim

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


A new canonical Hopf algebra called the quantum pseudo-Kähler plane is introduced. This quantum group can be viewed as a deformation quantization of the complex two-dimensional plane C2 with a pseudo-Kähler metric, or as a complexified version of the well-known quantum plane Hopf algebra. A natural class of nicely-behaved representations of the quantum pseudo-Kähler plane algebra is defined and studied, in the spirit of the previous joint work of the author and Frenkel. The tensor square of a unique irreducible representation decomposes into the direct integral of the irreducibles, and the unitary decomposition map is expressed by a special function called the modular double compact quantum dilogarithm, used in the recent joint work of the author and Scarinci on the quantization of 3d gravity for positive cosmological constant case. Then, from the associativity of the tensor cube, and from the maps between the left and the right duals, we construct unitary operators forming a new representation of Kashaev’s group of transformations of dotted ideal triangulations of punctured surfaces, as an analog of Kashaev’s quantum Teichmüller theory. The present work thus inspires one to look for a Kashaev-type quantization of 3d gravity for positive cosmological constant.

Original languageEnglish
Pages (from-to)2715-2763
Number of pages49
JournalCommunications in Mathematical Physics
Issue number3
StatePublished - Sep 2023

Bibliographical note

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.


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