  The main result of this paper is that the Weyl anomaly of a $z=2$
(1+1)-dimensional Lifshitz effective action can be derived from a
(2+1)-dimensional non-relativistic Schr$\mathrm{\ddot{o}}$dinger-invariant
Chern-Simons (NRSCS) action which was shown to be equivalent to a specific
Weyl-invariant non-projectable Horava-Lifshitz action of gravity. On a manifold
with a boundary, we will show that the (1+1)-dimensional Lifshitz Weyl anomaly
can be derived from a specific term, the torsional CS (tCS) term, in the NRSCS
action built from the gauge fields of the Weyl and special conformal symmetry
generators of the centrally-extended Schr$\mathrm{\ddot{o}}$dinger algebra. We
also focus on the $z=1$ Lifshitz Weyl anomaly and attempt to elicit its
geometric and physical nature, in particular its relationship with the Lorentz
anomaly of a (1+1)-dimensional CFT effective actions. We show that it is
directly related to the curvature scalar of the dual Lorentz connection, the
integral of which is known to be a topological invariant. We also point out
that making the anomalous Lifshitz quantum effective action Weyl-invariant
amounts to obtaining the equation of motion for a stationary chiral boson which
happens to be the spatial-component of the acceleration vector. By putting
boundary conditions on the spatial slices, the time dependence of the lapse
function in the Arnowitt, Deser and Misner (ADM) decomposition is eliminated
and the result is a Rindler metric. We finally discuss several issues related
to the (1+1)-dimensional Lifshitz Weyl anomaly regarding edge physics of
fractional quantum Hall states and anomaly cancellation by anomaly inflow.
