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Homework Help: Orthosymplectic super Lie Algebra OSp(4|4)

  1. Jun 18, 2014 #1
    1. The problem statement, all variables and given/known data

    We have 16 Bosonic Generators

    [tex] E_{ij} = - E_{ji}; i,j = 1 \dots 4 [/tex]
    [tex] E_{\bar{i} \bar{j}} = E_{\bar{i} \bar{j}}; \bar{i}, \bar{j} = 5 \dots 8 [/tex]

    and 16 fermionic generators

    [tex] E_{j \bar{j}}; j = 1 \dots 4, \bar{j} = 5 \dots 8 [/tex]

    that span the Lie superalgebra OSp(4|4).

    By the Lie-bracket we define the following map:

    [tex] Q(E) \equiv [E_{15},E]_{\pm} \in OSp(4|4). [/tex]

    + is the anticommutator and - the commutator.

    Determine the Kernel of Q [itex]Ker_Q.[/itex]

    2. Relevant equations
    We define the following matrix

    [tex] G_{IJ} = \begin{pmatrix} 0 & \mathbb{I}_2 & 0 & 0\\ \mathbb{I}_2 & 0 & 0 & 0 \\ 0 & 0 & 0 & \mathbb{I}_2\\ 0 & 0 & - \mathbb{I}_2 & 0 \end{pmatrix} [/tex]

    then the generators of OSp(4|4) are given by:

    [tex] E_{ij} = G_{ik}e_{kj} - G{jk}e_{ki}[/tex]

    [tex] E_{\bar{i} \bar{j}} = G_{\bar{i} \bar{k}}e_{\bar{k} \bar{j}} - G_{\bar{j}\bar{k}}e_{\bar{k}\bar{i}}[/tex]

    [tex] E_{i\bar{j}} = G_{ik}e_{k\bar{j}}[/tex]

    where [itex] (e_{IJ})_{KL} = \delta_{IL} \delta_{JK}. [/itex]

    The anti-/commutation relations are given by:

    [tex] [E_{i \bar{j}}, E_{\bar{k} \bar{l}}] = - G_{\bar{j} \bar{k}} E_{i \bar{l}} - G_{\bar{j} \bar{l}} E_{i \bar{k}} [/tex]

    [tex] \{ E_{i \bar{j}}, E_{k \bar{l}} \} = G_{i k} E_{ \bar{j}\bar{l}} - G_{\bar{j} \bar{l}} E_{i k} [/tex]

    3. The attempt at a solution

    [tex] Ker_Q = \{ E \in OSp(4|4) | Q(E) =0 \}. [/tex]


    [tex] [E_{15}, E_{\bar{k} \bar{l}}] = - G_{5 \bar{k}} E_{1 \bar{l}} - G_{5 \bar{l}} E_{1 \bar{k}} = 0[/tex]

    [tex] \{ E_{15}, E_{k \bar{l}} \} = G_{1 k} E_{ 5 \bar{l}} - G_{5 \bar{l}} E_{1 k} =0[/tex]

    I tried to calculate every generator E and then verify that the (anti-)commutator equals zero but the calculations are too long.

    Inserting the expressions for E into the (anti)-commutator relations and calculating [itex] G G [/itex] didn't get me far either.

    I have a commutator with many unknown generators and one known generator, how can I determine the unknown generators? I do not have any idea how to calculate this.

    Any help would be greatly appreciated!
  2. jcsd
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