On this page:
1 Some facts about
2 Case

The global symmetry algebra is . Let us for now neglect the reality structure, and just study the complex Lie superalgebra .

    1 Some facts about

    2 Case

1 Some facts about

We will start by making some observations about . This can be realized as supermatices acting in . Let us introduce additional structure: a pair of nondegenerate antisymmetric forms:

To abuse notations, we will denote them with the same letter . Such a form establishes an isomorphism of associative algebras:



In other words:

Then, the following map:


is a -automorphism of .

Elements of grading 3 and 1 are:

We observe:

Properties of trace:

2 Case

Without loss of generality we can assume:

Elements of grade 2 can be parametrized explicitly:


and similar formula for

These matrices have an interesting property:





and in particular: