  The double-scaled limit of the Sachdev-Ye-Kitaev (SYK) model takes the number
of fermions and their interaction number to infinity in a coordinated way. In
this limit, two entangled copies of the SYK model have a bulk description of
sorts known as the "chord Hilbert space." We analyze a symmetry algebra acting
on this Hilbert space, generated by the two Hamiltonians together with a
two-sided operator known as the chord number. This algebra is a deformation of
the JT gravitational algebra, and it contains a subalgebra that is a
deformation of the $\mathfrak{sl}_2$ near-horizon symmetries. The subalgebra
has finite-dimensional unitary representations corresponding to matter moving
around in a discrete Einstein-Rosen bridge. In a semiclassical limit the
discreteness disappears and the subalgebra simplifies to $\mathfrak{sl}_2$, but
with a non-standard action on the boundary time coordinate. One can make the
action of $\mathfrak{sl}_2$ algebra more standard at the cost of extending the
boundary circle to include some "fake" portions. Such fake portions also
accommodate certain subtle states that survive the semi-classical limit,
despite oscillating on the scale of discreteness. We discuss applications of
this algebra, including sub-maximal chaos, the traversable wormhole protocol,
and a two-sided OPE.
