# SU(3)-invariant Heisenberg XXX chain

1. Oct 20, 2014

### Maybe_Memorie

I'm studying the SU(3) invariant XXX chain as part of my Bachelor's thesis.

The monodromy matrix of this system can be written as a 3x3 matrix. We perform a 2x2 decomposition of it and write is as $T(\mu)=\left( \begin{array}{cc} A(\mu) & B(\mu) \\ C(\mu) & D (\mu) \end{array} \right)$

For a system to be integrable it has an R-matrix which satisfies the Yang-Baxter relation. I don't know how to decompose the R-matrix though.

Kulish/Resithiken writes it as, for the GL(N) case,
$R(\mu)=\left( \begin{array}{cccc} \mu & 0 & 0 & 0 \\ 0 & \mu I & I & 0 \\ 0 & I & \mu I & 0 \\ 0 & 0 & 0 & S(\mu) \end{array} \right)$
where I is the 2x2 identity matrix and S(u) is the SU(2) R-matrix.

The main reason this is confusing me is because when we write the Identity matrix and S(u) in the blocks we dont get the correct R-matrix, so i suspect there's a change of basis going on or something.

Any help is appreciated.

2. Oct 26, 2014