Finite GK-Dimensional pre-Nichols algebras of super and standard type

TL;DR

This paper classifies finite Gelfand-Kirillov dimension pre-Nichols algebras of super and standard types, showing most are quotients of distinguished pre-Nichols algebras with specific exceptions and constructing new examples via extensions.

Contribution

It establishes a classification of finite GK-dimensional pre-Nichols algebras of super and standard types, identifying exceptions and providing explicit constructions for them.

Findings

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

Explicit constructions of substitutes for exceptional cases as braided central extensions.

New finite GK-dimensional Hopf algebras obtained via bosonization.

Abstract

We prove that finite GK-dimensional pre-Nichols algebras of super and standard type are quotients of the corresponding distinguished pre-Nichols algebras, except when the braiding matrix is of type super A and the dimension of the braided vector space is three. For these two exceptions we explicitly construct substitutes as braided central extensions of the corresponding pre-Nichols algebras by a polynomial ring in one variable. Via bosonization this gives new examples of finite GK-dimensional Hopf algebras.