  A new canonical Hopf algebra called the quantum pseudo-K\"ahler plane is
introduced. This quantum group can be viewed as a deformation quantization of
the complex two-dimensional plane $\mathbb{C}^2$ with a pseudo-K\"ahler 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-K\"ahler
plane algebra is defined and studied, in the spirit of the previous joint work
of the author and I. B. 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 C. 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
Teichm\"uller theory. The present work thus inspires one to look for a
Kashaev-type quantization of 3d gravity for positive cosmological constant.
