V.Fock
Doubles

For a Riemann surface with boundary one can associate its double --- a closed
surface --- by glueing another copy of the surface along the boundary.
Teichmueller space of the double obviously Poisson projects onto the
Teichmueller space of the original surface. For a Poisson-Lie group one can
associate the so-called Heisenberg double --- a symplectic manifold of the
dimension twice the dimension of the group with Poisson projection to the
group. We shall see that these two (and some other) constructions are
particular cases of a unified construction of a cluster double.  We will show
also that the quantization of a double is even simpler that any other
quantization procedure of cluster varieties.