Finite GK-dimensional pre-Nichols algebras of (super)modular and unidentified type

TL;DR

This paper classifies finite GK-dimensional pre-Nichols algebras of diagonal type with connected diagrams, showing most are quotients of a distinguished algebra, and introduces specific substitutes for two exceptional cases.

Contribution

It completes the classification of finite GK-dimensional pre-Nichols algebras of diagonal type with connected diagrams, identifying exceptions and constructing substitutes.

Findings

Most finite GK-dimensional pre-Nichols algebras are quotients of a distinguished algebra.

Two exceptional cases are replaced by central extensions with explicit generators and relations.

The work provides a basis and detailed algebraic descriptions for these substitutes.

Abstract

We show that every finite GK-dimensional pre-Nichols algebra for braidings of diagonal type with connected diagram of modular, supermodular or unidentified type is a quotient of the distinguished pre-Nichols algebra introduced by the first-named author, up to two exceptions. For both of these exceptional cases, we provide a pre-Nichols algebra that substitutes the distinguished one in the sense that it projects onto all finite GK-dimensional pre-Nichols algebras. We build these two substitutes as non-trivial central extensions with finite GK-dimension of the corresponding distinguished pre-Nichols algebra. We describe these algebras by generators and relations, and provide a basis. This work essentially completes the study of eminent pre-Nichols algebras of diagonal type with connected diagram and finite-dimensional Nichols algebra.