Pointed Hopf algebras with standard braiding are generated in degree one

TL;DR

This paper proves that finite-dimensional pointed Hopf algebras with standard braiding are generated in degree one, confirming a long-standing conjecture and analyzing quantum Serre relations in this context.

Contribution

It establishes that such Hopf algebras are generated by group-like and skew-primitive elements, and details the behavior of quantum Serre relations in these algebras.

Findings

Pointed Hopf algebras with standard braiding are generated in degree one.

Quantum Serre relations hold in finite-dimensional coradically graded cases.

The paper clarifies how quantum Serre relations are lifted in the standard case.

Abstract

We show that any finite-dimensional pointed Hopf algebra over an abelian group $\Gamma$ such that its infinitesimal braiding is of standard type is generated by group-like and skew-primitive elements. This fact agrees with the long-standing conjecture by Andruskiewitsch and Schneider. We also show that the quantum Serre relations hold in any coradically graded pointed Hopf algebra over $\Gamma$ of finite dimension and determine how these relations are lifted in the standard case.