# Quantum supergroups VI. Roots of $1$

## TL;DR

This paper extends the theory of quantum groups to quantum covering groups with parameters q and pi, establishing key homomorphisms and tensor product theorems at roots of unity, unifying quantum groups and supergroups.

## Contribution

It introduces Frobenius-Lusztig homomorphism and Lusztig-Steinberg tensor product theorem for quantum covering groups at roots of 1, generalizing existing results for quantum groups.

## Key findings

- Established Frobenius-Lusztig homomorphism for quantum covering groups.
- Proved Lusztig-Steinberg tensor product theorem in this setting.
- Specialization at pi=1 recovers classical quantum group results.

## Abstract

A quantum covering group is an algebra with parameters $q$ and $\pi$ subject to $\pi^2=1$ and it admits an integral form; it specializes to the usual quantum group at $\pi=1$ and to a quantum supergroup of anisotropic type at $\pi=-1$. In this paper we establish the Frobenius-Lusztig homomorphism and Lusztig-Steinberg tensor product theorem in the setting of quantum covering groups at roots of 1. The specialization of these constructions at $\pi=1$ recovers Lusztig's constructions for quantum groups at roots of 1.