 Solution

For the unheaded extended-algebra request, fix the central-charge normalization explicitly:The central charges are scalar numbers on an irreducible massive representation. A unitary skew-diagonalization of an antisymmetric matrix puts $Z$ into two-by-two blocks $\left(\begin{array}{cc}0& z_r\\ -z_r& 0\end{array}\right)$ and, for odd $\mathcal{N}$, a leftover zero entry. Since this change of supercharge basis is unitary, it preserves the mixed anticommutator. Go to the rest frame $P^{\mu }=(M,0,0,0)$.

For one block set $q_{\alpha }=Q_{\alpha }^1$ and $r_{\alpha }={ϵ}_{\alpha \gamma }(Q_{\gamma }^2{)}^{†}$, with ${ϵ}_{12}=1$. Direct use of the algebra givesWrite $z=|z|e^{i\phi }$ and form the combinations suggested by the hint:ThenEach diagonal anticommutator has nonnegative expectation because it is the sum of squared norms of an operator and its adjoint. Therefore $M-|z_r|\ge 0$ for every block, provingIf the algebra is instead written with $\{Q,Q\}=ϵZ$ without the factor two, the same proof gives $M\ge \frac{1}{2}{\max }_r|Z_r|$. This is the normalization used by some definitions of the BPS bound in supersymmetry; the physical bound is unchanged by the naming of the central charge.

A massive BPS state saturates at least one block bound. At saturation $\{B_{\alpha },B_{\alpha }^{†}\}=0$, so both $B_{\alpha }$ and $B_{\alpha }^{†}$ annihilate every state in its multiplet. The two complex inactive oscillators are four preserved real supersymmetries per saturated block. If $r$ blocks saturate, the preserved fraction is $r/\mathcal{N}$ and only $2\mathcal{N}-2r$ complex fermionic oscillators remain active. For a spin-$j$ Clifford vacuum,compared with $(2j+1)2^{2\mathcal{N}}$ for a long massive multiplet. Thus BPS states have an exact mass-charge relation, preserve part of the supersymmetry, and belong to shortened representations. A physical CPT-invariant spectrum may additionally require the conjugate-charge multiplet. The short representation protects the saturated mass relation while it exists; it does not prohibit a decay at a threshold where central-charge phases align and the sum of daughter masses equals the bound.