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

1. Jun 18, 2014

### silverwhale

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

We have 16 Bosonic Generators

$$E_{ij} = - E_{ji}; i,j = 1 \dots 4$$
$$E_{\bar{i} \bar{j}} = E_{\bar{i} \bar{j}}; \bar{i}, \bar{j} = 5 \dots 8$$

and 16 fermionic generators

$$E_{j \bar{j}}; j = 1 \dots 4, \bar{j} = 5 \dots 8$$

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

By the Lie-bracket we define the following map:

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

+ is the anticommutator and - the commutator.

Determine the Kernel of Q $Ker_Q.$

2. Relevant equations
We define the following matrix

$$G_{IJ} = \begin{pmatrix} 0 & \mathbb{I}_2 & 0 & 0\\ \mathbb{I}_2 & 0 & 0 & 0 \\ 0 & 0 & 0 & \mathbb{I}_2\\ 0 & 0 & - \mathbb{I}_2 & 0 \end{pmatrix}$$

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

$$E_{ij} = G_{ik}e_{kj} - G{jk}e_{ki}$$

$$E_{\bar{i} \bar{j}} = G_{\bar{i} \bar{k}}e_{\bar{k} \bar{j}} - G_{\bar{j}\bar{k}}e_{\bar{k}\bar{i}}$$

$$E_{i\bar{j}} = G_{ik}e_{k\bar{j}}$$

where $(e_{IJ})_{KL} = \delta_{IL} \delta_{JK}.$

The anti-/commutation relations are given by:

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

$$\{ E_{i \bar{j}}, E_{k \bar{l}} \} = G_{i k} E_{ \bar{j}\bar{l}} - G_{\bar{j} \bar{l}} E_{i k}$$

3. The attempt at a solution

$$Ker_Q = \{ E \in OSp(4|4) | Q(E) =0 \}.$$

Thus,

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

$$\{ E_{15}, E_{k \bar{l}} \} = G_{1 k} E_{ 5 \bar{l}} - G_{5 \bar{l}} E_{1 k} =0$$

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 $G G$ 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!