Orthosymplectic super Lie Algebra OSp(4|4)

[silverwhale]

Homework Statement

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 $Ker_Q.$

Homework 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:

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

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

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!

 

[mark726]

One approach to solving this problem is to use the properties of the Lie superalgebra OSp(4|4) to simplify the calculations. For example, you can use the fact that the generators E_{ij} and E_{\bar{i} \bar{j}} are antisymmetric and the generators E_{i \bar{j}} are symmetric. This can help to reduce the number of terms in the (anti-)commutator relations and make the calculations more manageable.

Another approach is to use the matrix representation of the generators and the Lie superalgebra. This can help to simplify the calculations by allowing you to work with matrices instead of individual generators. For example, you can use the matrix representation of the generators to write the (anti-)commutator relations in terms of matrix multiplication and use the properties of matrices to simplify the expressions.

You can also try to use the fact that the generators E_{ij}, E_{\bar{i} \bar{j}}, and E_{i \bar{j}} form a basis for the Lie superalgebra OSp(4|4). This means that any element in the Lie superalgebra can be written as a linear combination of these generators. Using this fact, you can try to express the unknown generators in terms of the known ones and then use the (anti-)commutator relations to solve for the coefficients.

Finally, you can also try to use computer algebra systems to help with the calculations. These systems can handle large expressions and can help to simplify them using various algebraic manipulations. They can also help to verify your results and check for any errors in your calculations.