  Based on the Sloan Digital Sky Survey Data Release 16, we have detected the
large-scale structure of Ly$\alpha$ emission in the Universe at redshifts $z =
2$--3.5 by cross-correlating quasar positions and Ly$\alpha$ emission imprinted
in the residual spectra of luminous red galaxies. We apply an analytical model
to fit the corresponding Ly$\alpha$ surface brightness profile and multipoles
of the redshift-space quasar-Ly$\alpha$ emission cross-correlation function.
The model suggests an average cosmic Ly$\alpha$ luminosity density of
${6.6_{-3.1}^{+3.3}}\times 10^{40} {\rm erg\, s^{-1} cMpc^{-3}}$, a $\sim
2\sigma$ detection with a median value about 8--9 times those estimated from
deep narrowband surveys of Ly$\alpha$ emitters at similar redshifts. Although
the low signal-to-noise ratio prevents us from a significant detection of the
Ly$\alpha$ forest-Ly$\alpha$ emission cross-correlation, the measurement is
consistent with the prediction of our best-fit model from quasar-Ly$\alpha$
emission cross-correlation within current uncertainties. We rule out the
scenario that these Ly$\alpha$ photons mainly originate from quasars. We find
that Ly$\alpha$ emission from star-forming galaxies, including contributions
from that concentrated around the galaxy centers and that in the diffuse
Ly$\alpha$ emitting halos, is able to explain the bulk of the the Ly$\alpha$
luminosity density inferred from our measurements. Ongoing and future surveys
can further improve the measurements and advance our understanding of the
cosmic Ly$\alpha$ emission field.
