Closed-form solutions of the Wheeler-DeWitt equation in a scalar-vector
  field cosmological model by Lie symmetries
  We apply as selection rule to determine the unknown functions of a
cosmological model the existence of Lie point symmetries for the Wheeler-DeWitt
equation of quantum gravity. Our cosmological setting consists of a flat
Friedmann-Robertson-Walker metric having the scale factor $a(t)$, a scalar
field with potential function $V(\phi)$ minimally coupled to gravity and a
vector field of its kinetic energy is coupled with the scalar field by a
coupling function $f(\phi)$. Then, the Lie symmetries of this dynamical system
are investigated by utilizing the behavior of the corresponding minisuperspace
under the infinitesimal generator of the desired symmetries. It is shown that
by applying the Lie symmetry condition the form of the coupling function and
also the scalar field potential function may be explicitly determined so that
we are able to solve the Wheeler-DeWitt equation. Finally, we show how we can
use the Lie symmetries in order to construct conservation laws and exact
solutions for the field equations.
