Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VII. Semisimple classes in PSL(n,q) and PSp(2n,q)

TL;DR

This paper classifies finite-dimensional pointed Hopf algebras over certain finite simple groups of Lie type, showing most have trivial structure except in specific low-rank cases, by analyzing Nichols algebras over semisimple classes.

Contribution

It proves that Nichols algebras over semisimple orbits in PSL(n,q) and PSp(2n,q) are infinite-dimensional unless in low-rank cases, leading to classification of associated Hopf algebras.

Findings

Most Nichols algebras over semisimple classes are infinite-dimensional.

Finite-dimensional pointed Hopf algebras over PSL(n,q) and PSp(2n,q) are group algebras.

Complete classification of such Hopf algebras for specified groups.

Abstract

We show that the Nichols algebra of a simple Yetter-Drinfeld module over a projective special linear group over a finite field whose support is a semisimple orbit has infinite dimension, provided that the elements of the orbit are reducible; we obtain a similar result for all semisimple orbits in a finite symplectic group except in low rank. We prove that orbits of irreducible elements in the projective special linear groups could not be treated with our methods. We conclude that any finite-dimensional pointed Hopf algebra H with group of grouplike elements isomorphic to PSL(n,q) (n greater than or equal to 4), PSL(3,q) (q greater than 2), or PSp(2n,q) (n greater than or equal to 3), is isomorphic to a group algebra, completing work in arXiv:1506.06794.