Symmetries

Symmetries

The global symmetry algebra is

. Let us for now neglect the reality structure, and just study the complex Lie superalgebra

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:

(1)

where

In other words:

Then, the following map:

(2)

is a

-automorphism of

Elements of grading 3 and 1 are:

We observe:

Properties of trace:

Without loss of generality we can assume:

Elements of grade 2 can be parametrized explicitly:

where

and similar formula for

These matrices have an interesting property:

Therefore:

(3)

where

(4)

and in particular:

(5)