  We perform a forecast analysis of the ability of the LISA space-based
interferometer to reconstruct the dark sector interaction using gravitational
wave (GW) standard sirens at high redshift. We employ Gaussian process methods
to reconstruct the distance-redshift relation in a model independent way. We
adopt simulated catalogues of standard sirens given by merging massive black
hole binaries (MBHBs) visible by LISA, with an electromagnetic (EM) counterpart
detectable by future telescopes. The catalogues are constructed considering
three different astrophysical scenarios for the evolution of MBHB mergers based
on the semi-analytic model of E. Barausse (2012). We first use these standard
siren datasets to assess the potential of LISA in reconstructing a possible
interaction between vacuum dark energy and dark matter. Then we combine the
LISA cosmological data with supernovae data simulated for the Dark Energy
Survey (DES). We consider two scenarios distinguished by the time duration of
the LISA mission: 5 and 10 years. Using only LISA standard siren data, the dark
sector interaction can be well reconstructed from redshift $z\sim1$ to $z\sim3$
(5 yr) and $z\sim1$ up to $z\sim5$ (10 yr), though the reconstruction is
inefficient at lower redshift. When combined with the DES datasets, the
interaction is well reconstructed in the whole redshift region from $z\sim0$ to
$z\sim3$ (5 yr) and $z\sim0$ to $z\sim5$ (10 yr). MBHB standard sirens can thus
be used to constrain the dark sector interaction at redshift ranges not
reachable by usual supernovae datasets which probe only the $z\lesssim 1.5$
range. GW standard sirens will not only constitute a complementary and
alternative way, with respect to familiar EM observations, to probe the cosmic
expansion, but will also provide new tests to constrain possible deviations
from the standard $\Lambda$CDM dynamics, especially at high redshift.
