1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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]

    Thus,

    [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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Orthosymplectic super Lie Algebra OSp(4|4)
  1. \phi^4 CFT question (Replies: 0)

  2. Identify the 4-force (Replies: 0)

Loading...