supersymmetry

# Contents

## Idea

In discussion of supersymmetry: the number of generators of odd degree, suitably conceived.

## Definition

A supersymmetry super Lie algebra or super Lie group is determined by the underlying bosonic algebra/group (body) and a real spin representation $\mathbf{N}$.

One says that the corresponding number of supersymmetries is either the dimension

$N \;\coloneqq\; dim_{\mathbb{R}}\big( \mathbf{N} \big)$

of the real spin representation $\mathbf{N}$, and as such denoted by a roman “$N$”,

or, alternatively, the multiplicity

$\mathcal{N} \;\coloneqq\; (n_i)_{i \in I}$

of the irreducible real spin representations $\mathbf{N}^{irr}_{i}$ in a direct sum decomposition

$\mathbf{N} \;\simeq\; \underset{i \in I}{\oplus} n_i \mathbf{N}^{irr}_i$

of this real spin representation $\mathbf{N}$, and as such denoted by a list of calligraphic “$\mathcal{N}$”s.

Typically there is either a single irrep or precisely two, in which case these multiplicities are either a single natural number

$\mathcal{N} \;\in\; \mathbb{N}$

or a pair of them

$\mathcal{N} \;=\; (\mathcal{N}_+, \mathcal{N}_-) \,,$

respectively.

## References

See the references at supersymmetry, for instance