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

Research paper by **K. A. Dancer**

Indexed on: **16 Nov '05**Published on: **16 Nov '05**Published in: **Mathematics - Quantum Algebra**

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#### Abstract

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.