# Yang baxter equation question

1. Oct 27, 2011

### wik_chick88

1. The problem statement, all variables and given/known data

show that the following matrix satisfies the Yang-Baxter equation

2. Relevant equations

$R(u) = (1 - u) (E^1_1 \otimes E^1_1 + E^2_2 \otimes E^2_2) - u (E^1_1 \otimes E^2_2 + E^2_2 \otimes E^1_1) + E^1_2 \otimes E^2_1 + E^2_1 \otimes E^1_2$

3. The attempt at a solution

the yang-baxter equation is:
$R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}$
where $R = a_i \otimes b_i$
and $R_{12} = a_i \otimes b_i \otimes I, R_{13} = a_i \otimes I \otimes b_i, R_{23} = \otimes I \otimes a_i \otimes b_i$
and $E^i_j$ are the usual 2 x 2 elementary matrices

i really don't know how to start this question. do i have to somehow represent R(u) in the form $a_i \otimes b_i$? i also read somewhere (not in our notes) that $R_{12}$ is a function of u, $R_{13}$ is a function of (u + v) and $R_{23}$ is a function of (v)...is this correct and does it matter for this question?
before i realised that $\otimes$ was the tensor product and not just the matrix multiplication operation, I simplified R(u) right down to $R(u) = (2-u) I$ by expanding the matrices...does this help in any way?