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We describe the Lorentzian version of the Kapovitch-Millson phase space forpolyhedra with $N$ faces. Starting with the Schwinger representation of the$\mathfrak{su}(1,1)$ Lie algebra in terms of a pair of complex variables (orspinor), we define the phase space for a space-like vectors in thethree-dimensional Minkowski space $\mathbb{R}^{1,2}$. Considering $N$ copies ofthis space, quotiented by a closure constraint forcing the sum of those3-vectors to vanish, we obtain the phase space for Lorentzian polyhedra with$N$ faces whose normal vectors are space-like, up to Lorentz transformations.We identify a generating set of $SU(1,1)$-invariant observables, whose flow bythe Poisson bracket generate both area-preserving and area-changingdeformations. We further show that the area-preserving observables form a$\mathfrak{gl}_{N}(\mathbb{R})$ Lie algebra and that they generate a$GL_{N}(\mathbb{R})$ action on Lorentzian polyhedra at fixed total area. Thataction is cyclic and all Lorentzian polyhedra can be obtained from a totallysquashed polyhedron (with only two non-trivial faces) by a $GL_{N}(\mathbb{R})$transformation. All those features carry on to the quantum level, where quantumLorentzian polyhedra are defined as $SU(1,1)$ intertwiners between unitary$SU(1,1)$-representations from the principal continuous series. Those$SU(1,1)$-intertwiners are the building blocks of spin network states in loopquantum gravity in 3+1 dimensions for time-like slicing and the presentanalysis applies to deformations of the quantum geometry of time-likeboundaries in quantum gravity, which is especially relevant to the study ofquasi-local observables and holographic duality.

Deformations of Lorentzian Polyhedra: Kapovich-Millson phase space and SU(1,1) Intertwiners