Algebras of Non-Local Screenings and Diagonal Nichols Algebras

Abstract

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.

References in corpus (5)

• Logarithmic CFTs connected with simple Lie algebras
• Modularization of small quantum groups
• Quantum SL(2) and logarithmic vertex operator algebras at (p,1)-central charge
• Characterizing braided tensor categories associated to logarithmic vertex operator algebras
• Multiparameter quantum groups at roots of unity