  The Kohn-Luttinger mechanism for unconventional superconductivity (SC) driven
by weak repulsive electron-electron interactions on a periodic lattice is
generalized to the quasicrystal (QC) via a real-space perturbative approach.
The repulsive Hubbard model on the Penrose lattice is studied as an example, on
which a classification of the pairing symmetries is performed and a pairing
phase diagram is obtained. Two remarkable properties of these pairing states
are revealed, due to the combination of the presence of the point-group
symmetry and the lack of translation symmetry on this lattice. Firstly, the
spin and spacial angular momenta of a Cooper pair is de-correlated: for each
pairing symmetry, both spin-singlet and spin-triplet pairings are possible even
in the weak-pairing limit. Secondly, the pairing states belonging to the 2D
irreducible representations of the $D_5$ point group can be
time-reversal-symmetry-breaking topological SCs carrying spontaneous bulk super
current and spontaneous vortices. These two remarkable properties are general
for the SCs on all QCs, and are rare on periodic lattices. Our work starts the
new area of unconventional SCs driven by repulsive interactions on the QC.
