# The Jacobian, reflection arrangement and discriminant for reflection Hopf algebras

## TL;DR

This paper introduces noncommutative analogs of classical algebraic invariants like the Jacobian, reflection arrangement, and discriminant for actions of finite-dimensional semisimple Hopf algebras on regular algebras, expanding the understanding of symmetry in noncommutative geometry.

## Contribution

It defines and studies the Jacobian, reflection arrangement, and discriminant in the context of noncommutative algebra, specifically for Hopf algebra actions on Artin-Schelter regular algebras.

## Key findings

- Defined noncommutative Jacobian, reflection arrangement, and discriminant.
- Extended classical invariants to noncommutative setting.
- Provided foundational framework for symmetry analysis in noncommutative algebra.

## Abstract

We study finite dimensional semisimple Hopf algebra actions on noetherian connected graded Artin-Schelter regular algebras, and introduce definitions of the Jacobian, the reflection arrangement and the discriminant in a noncommutative setting.