Hopf actions on vertex operator algebras

TL;DR

This paper studies how Hopf algebras act on vertex operator algebras, showing under certain conditions that such actions are essentially group actions, with implications for symmetry and structure.

Contribution

It establishes conditions under which Hopf actions on vertex operator algebras reduce to group actions, extending understanding of symmetry in algebraic structures.

Findings

Semisimple Hopf actions admit a Schur-Weyl type decomposition.

Faithful finite-dimensional Hopf actions are group actions.

Semisimple, inner faithful actions are equivalent to group actions.

Abstract

The Hopf actions on vertex operator algebras are investigated. If the action is semisimple, a Schur-Weyl type decomposition is obtained. When the Hopf algebra is finite dimensional and the action is faithful, the action is a group action. Moreover if the Hopf algebra is finite dimensional and the action is semisimple and inner faithful, the action is also a group action. In this case, inner faithfulness is equivalent to faithfulness.