Minkowski spacetime In Minkowski spacetime, the Haag–Łopuszański–Sohnius theorem states that the SUSY algebra with N spinor generators is as follows. The even part of the star Lie superalgebra is the direct sum of the Poincaré algebra and a reductive Lie algebra B (such that its self-adjoint part is the tangent space of a real compact Lie group). The odd part of the algebra would be where and are specific representations of the Poincaré algebra. (Compared to the notation used earlier in the article, these correspond and , respectively, also see the footnote where the previous notation was introduced). Both components are conjugate to each other under the * conjugation. V is an N-dimensional complex representation of B and V* is its dual representation. The Lie bracket for the odd part is given by a symmetric equivariant pairing {.,.} on the odd part with values in the even part. In particular, its reduced intertwiner from to the ideal of the Poincaré algebra generated by translations is given as the product of a nonzero intertwiner from to (1/2,1/2) by the "contraction intertwiner" from to the trivial representation. On the other hand, its reduced intertwiner from is the product of a (antisymmetric) intertwiner from to (0,0) and an antisymmetric intertwiner A from to B. Conjugate it to get the corresponding case for the other half. N = 1 B is now (called R-symmetry) and V is the 1D representation of with charge 1. A (the intertwiner defined above) would have to be zero since it is antisymmetric. Actually, there are two versions of N=1 SUSY, one without the (i.e. B is zero-dimensional) and the other with . N = 2 B is now and V is the 2D doublet representation of with a zero charge. Now, A is a nonzero intertwiner to the part of B. Alternatively, V could be a 2D doublet with a nonzero charge. In this case, A would have to be zero. Yet another possibility would be to let B be . V is invariant under and and decomposes into a 1D rep with charge 1 and another 1D rep with charge -1. The intertwiner A would be complex with the real part mapping to and the imaginary part mapping to . Or we could have B being with V being the doublet rep of with zero charges and A being a complex intertwiner with the real part mapping