Algebras of Non-Local Screenings and Diagonal Nichols Algebras

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 satisfy relations quantum shuffle or Nichols diagonal braiding, which encodes non-locality and non-integrality. present article, take all finite-dimensional algebras, as classified by Heckenberger, find lattice realizations braiding that are compatible with reflections. Usually, unique come one- two-parameter families. Examples include Lie superalgebras. then study screenings improved methods. Typically, for positive definite lattices obtain algebra, such part group, negative extension generalizing infinite group large center.