Definition of the Lie algebraWe will start with . We define the Lie algebra as a sum of two copies of , plus some finite-dimensional Lie algebra (as linear spaces):which are all “glued together” as follows:

where is the structure constants of —the algebra of supersymmetries of . Comments:

Our is not a quadratic algebra (because of Eqs. (A), (B) and (C))

Eq. (*) defines a quadratic subalgebra which we call , and Eq. (**) defines

Each and is the same as the Yang-Mills (or Maxwell) algebra (20)

The rest of relations tell us how to “glue together” and

Eq. (C) tells us that the generators form —

rotations around a point in