  The proposed electron-proton collider LHeC is a unique facility where
electroweak interactions can be studied with a very high precision in a largely
unexplored kinematic regime of spacelike momentum transfer. We have simulated
inclusive neutral- and charged-current deep-inelastic lepton proton scattering
cross section data at center-of-mass energies of 1.2 and 1.3 TeV including
their systematic uncertainties. Based on simultaneous fits of electroweak
physics parameters and parton distribution functions, we estimate the
uncertainties of Standard Model parameters as well as a number of parameters
describing physics beyond the Standard Model, for instance the oblique
parameters $S$, $T$, and $U$. An unprecedented precision at the sub-percent
level is expected for the measurement of the weak neutral-current couplings of
the light-quarks to the $Z$ boson, $g_{A/V}^{u/d}$, improving their present
precision by more than an order of magnitude. The weak mixing angle can be
determined with a precision of about $\Delta \sin^2\theta_\textrm{W} = \pm
0.00015$, and its scale dependence can be studied in the range between about 25
and 700 GeV. An indirect determination of the $W$-boson mass in the on-shell
scheme is possible with an experimental uncertainty down to $\Delta
m_{W}=\pm6\,\textrm{MeV}$. We discuss how the uncertainties of such
measurements in deep-inelastic scattering compare with those from measurements
in the timelike domain, e.g. at the $Z$-pole, and which aspects of the
electroweak interaction are unique to measurements at the LHeC, for instance
electroweak parameters in charged-current interactions. We conclude that the
LHeC will determine electroweak physics parameters, in the spacelike region,
with unprecedented precision leading to thorough tests of the Standard Model
and possibly beyond.
