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Subspace

We will now study the necessary properties of the previously defined space .

Let us define the bracket:

(23)

which satisfies . Because of Eq. (22), we will impose the following additional requirement:

(24)

Moreover, Eq. (21) implies the existence of a map

such that:

(25)

We can prove that:

Under the conditions (25) and (24) the space is a Lie superalgebra under

(i.e. satisfies the Jacobi identity).

There is a possibility to impose a slightly weaker condition. Instead of requiring (25) we can just ask for . And continue to require (24). This is the same as to say that:

  1. is closed under and

  2. is antisymmetric

(Because is constructed from (which already satisfies the Jacobi identity), the antisymmetry of already implies that satisfies the Jacobi identity.)

Therefore is a Lie algebra. There is a map . The image of this map, with flipped Grassmann parity, naturally denoted , is a Lie subalgebra of :

We will also request that:

the map is injective

In other words:

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