# Introduction

This chapter describes the functionality for quantum groups in Magma. First there are a few sections that briefly describe the theoretical background behind quantum groups (or, more precisely, quantized enveloping algebras). This fixes the notation and the terminology that we use (these vary somewhat in the literature). For this we mainly follow [[Jantzen, 1996](../../references.md#cite-jantzen96)]. In the remainder we describe the functions that exist in Magma for constructing and working with quantum groups and their representations.

In Magma, quantized enveloping algebras have type `AlgQUE` and their elements have type `AlgQUEElt`. These types inherit from `AlgPBW` and `AlgPBWElt` respectively, which are general types for algebras with a PBW basis and their elements and inherit from `GenMPolB`, `Alg` and `Rng` and their element types.