Algebras of Non-Local Screenings and Diagonal Nichols Algebras

TL;DR

This paper explores the algebraic structures arising from non-local screening operators in vertex algebras, classifying lattice realizations of diagonal Nichols algebras and analyzing their associated algebraic properties.

Contribution

It classifies all lattice realizations of finite-dimensional diagonal Nichols algebras compatible with reflections and studies their algebraic structures in detail.

Findings

Classified all lattice realizations of diagonal Nichols algebras compatible with reflections.

Identified realizations corresponding to Lie superalgebras.

Described algebraic structures for positive and negative definite lattices.

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.