The subject of quantum groups is a fascinating and new area of mathematics.
The term “quantum groups” was first introduced by Vladimir Drinfeld in the 1980s. There is no formal definition for the term “quantum groups”, but in general it stands for certain special Hopf algebras.
In some of the literature, a quantum group is defined as a noncommutative and noncocommutative Hopf algebra.
Quantum groups have close connections with many areas of mathematics and physics.
A starting approach is to study the theory of quantum groups, and related topics like Hopf algebras. An example can be as follows.
Chapter 1 introduces the definition of Hopf algebras and its properties.
Chapter 2 focuses on the theory of quantum groups. Chapter 3 includes examples of quantum groups, with some examples from physics.
The main reference for the project can be Quantum Groups by C. Kassel ().
For a project, one can refer to Part One of the book, from Chapter I to VII. Before researching on quantum groups, one can also study Hopf algebras, including viewing online lectures by F. Ardila ().
A good project topic is to study and write a summary and a brief introduction to the vast subject of quantum groups.