a restriction, the full ${{\mathfrak{psu}}}(2,2|4)$ algebra splits into three components. One component annihilates all the states in the subsector, and the second component maps states in the subsector out of the subsector. The third component, a subalgebra, acts within the subsector non-trivially, and gives the sector its name. For the case we consider in this work, this subalgebra is ${{\mathfrak{u}}}(1)^2 \ltimes ({{\mathfrak{psu}}}(1,1|2) \times {{\mathfrak{psu}}}(1|1)^2) \ltimes {{\mathfrak{u}}}(1)$. The ${{\mathfrak{u}}}(1)^2$ consists of two external automorphisms, the length $L$ and the hypercharge $B$. The ${{\mathfrak{u}}}(1)$, is the quantum correction to the dilatation generator, ${\delta{\mathfrak{D}}}$, which appears as the central charge for both ${{\mathfrak{psu}}}(1,1|2)$ and ${{\mathfrak{psu}}}(1|1)^2$.[^6] Furthermore, the ${{\mathfrak{psu}}}(1|1)^2$ acts trivially classically. Now we describe the restriction to this subsector and the corresponding symmetry algebra, and we present the leading non-vanishing actions of the algebra on the states of the subsector.

The restriction to the $ {{\mathfrak{su}}}(1,1|2)$ sector {#sec:restriction}
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To restrict to this sector, we must set the classical dimensions of states simultaneously equal to the following linear combinations of the eigenvalues of the Cartan generators of the $ {{\mathfrak{psu}}}(2,2|4)$ algebra [@Beisert:2004ry], $$\begin{aligned}
 \label{eq:restriction}
D_0 &=& s_1+\frac{1}{2}q_2+p+\frac{3}{2}q_1=s_2+\frac{1}{2}q_1+p+\frac{3}{2}q_2 \end{aligned}$$ Here $[q_1, \, p, \, q_2]$ are the Dynkin labels of the ${{\mathfrak{su}}}(4)$ subalgebra and 