Quantum Groups – a study of 𝑈𝑞(𝔰𝔩2)

This group project will explore the theory of quantum groups and their representations. We focus on quantum groups of Drinfeld–Jimbo type, that is, deformations of the universal enveloping algebras of certain Lie algebras. We introduce the aspects of the representation theory of 𝑈𝑞(𝔰𝔩2), looking at both generic and specific values of 𝑞. Then we consider a combinatorial formula using crystal bases to compute the decomposition of tensor products of particular 𝑈𝑞(𝔰𝔩𝑛)-modules. Finally, we discuss the quasitriangularity condition with a focus on the extra structure it provides to the category of 𝑈𝑞(𝔰𝔩2)-modules and its relations to exactly solvable lattice models in statistical mechanics.