Solutions to the Yang-Baxter Equation and Casimir Invariants for the Quantised Orthosymplectic Superalgebra

For the last fifteen years quantum superalgebras have been used to model supersymmetric quantum systems. A class of quasi-triangular Hopf superalgebras, they each contain a universal $R$-matrix, which automatically satisfies the Yang--Baxter equation. Applying the vector representation to the left-hand side of a universal $R$-matrix gives a Lax operator. These are of significant interest in mathematical physics as they provide solutions to the Yang--Baxter equation in an arbitrary representation, which give rise to integrable models. In this thesis a Lax operator is constructed for the quantised orthosymplectic superalgebra $U_q[osp(m|n)]$ for all $m > 2, n \geq 2$ where $n$ is even. This can then be used to find a solution to the Yang--Baxter equation in an arbitrary representation of $U_q[osp(m|n)]$, with the example of the vector representation given in detail. In studying the integrable models arising from solutions to the Yang--Baxter equation, it is desirable to understand the representation theory of the superalgebra. Finding the Casimir invariants of the system and exploring their behaviour helps in this understanding. In this thesis the Lax operator is used to construct an infinite family of Casimir invariants of $U_q[osp(m|n)]$ and to calculate their eigenvalues in an arbitrary irreducible representation.