On Pointed Hopf Algebras with Sporadic Simple Groups ${\rm HS}$ and ${\rm Co3}$

TL;DR

This paper classifies all quasi--1-type Nichols algebras over the sporadic simple groups HS and Co3, identifying which are finite-dimensional and thereby advancing the understanding of their algebraic structures.

Contribution

It provides a complete classification of quasi--1-type Nichols algebras over HS and Co3, revealing which are finite-dimensional, a novel result in the study of these groups.

Findings

All quasi--1-type Nichols algebras over HS and Co3 are identified.

Finite-dimensional quasi--1-type Nichols algebras are explicitly characterized.

Infinite-dimensional Nichols algebras are associated with non quasi--1-type cases.

Abstract

Every non quasi- -1-type Nichols algebra is infinite dimensional. All quasi- -1-type Nichols algebra over sporadic simple groups ${\rm HS}$ and ${\rm Co3}$ are found.