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Yang baxter equation question

  1. Oct 27, 2011 #1
    1. The problem statement, all variables and given/known data

    show that the following matrix satisfies the Yang-Baxter equation

    2. Relevant equations

    [itex]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[/itex]

    3. The attempt at a solution

    the yang-baxter equation is:
    [itex]R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}[/itex]
    where [itex]R = a_i \otimes b_i[/itex]
    and [itex]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[/itex]
    and [itex]E^i_j[/itex] 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 [itex]a_i \otimes b_i[/itex]? i also read somewhere (not in our notes) that [itex]R_{12}[/itex] is a function of u, [itex]R_{13}[/itex] is a function of (u + v) and [itex]R_{23}[/itex] is a function of (v)...is this correct and does it matter for this question?
    before i realised that [itex]\otimes[/itex] was the tensor product and not just the matrix multiplication operation, I simplified R(u) right down to [itex]R(u) = (2-u) I[/itex] by expanding the matrices...does this help in any way?
    someone please help!!
     
  2. jcsd
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