  In a vertex algebra setting, we consider non-local screening operators
associated to the basis of any non-integral lattice. We have previously shown
that, under certain restrictions, these screening operators satisfy the
relations of a quantum shuffle algebra or Nichols algebra associated to a
diagonal braiding, which encodes the non-locality and non-integrality. In the
present article, we take all finite-dimensional diagonal Nichols algebras, as
classified by Heckenberger, and find all lattice realizations of the braiding
that are compatible with reflections. Usually, the realizations are unique or
come as one- or two-parameter families. Examples include realizations of Lie
superalgebras. We then study the associated algebra of screenings with improved
methods. Typically, for positive definite lattices we obtain the Nichols
algebra, such as the positive part of the quantum group, and for negative
definite lattices we obtain a certain extension of the Nichols algebra
generalizing the infinite quantum group with a large center.
