A supersymmetric extension for ${\mathrm{AdS}}_4$ background was found by Konstein and Vasiliev in Nucl.Phys.B331:475-499,1990, and later generalised by Vasiliev in hep-th/0404124 to higher dimensions. In 4d, there are three classes of infinite-dimensional extended higher spin superalgebras which generate symmetries of the higher spin equations of motion on ${\mathrm{AdS}}_4$. In each case, the bosonic part contains a subalgebra of the form ${\mathfrak{so}}(3,2) \oplus {\mathfrak{g}}(m) \oplus {\mathfrak{g}}(n)$, comprising the ${\mathrm{AdS}}_4$ isometries and ${\mathfrak{g}}$ being either ${\mathfrak{u}}$, ${\mathfrak{o}}$ or ${\mathfrak{usp}}$. The corresponding higher spin superalgebras are denoted ${\mathfrak{hg}}(m,n|4)$. They contain the usual $N$-extended lie superalgebra ${\mathfrak{osp}}(N|4)$ as a subalgebra only when $m=n$. Indeed, for $m\neq n$, massless unitary irreps of ${\mathfrak{hg}}(m,n|4)$ contain a different number of bosons and fermions. In the simplest class with ${\mathfrak{g}}={\mathfrak{u}}$, bosons have all integer spins $\gt$ 1 and are in the adjoint of ${\mathfrak{u}}(m) \oplus {\mathfrak{u}}(n)$ while fermions have all half-integer spins $\gt$ 3/2 and are in the bifundamental of ${\mathfrak{u}}(m) \oplus {\mathfrak{u}}(n)$. (The standard spin 2 graviton is contained in a diagonal ${\mathfrak{u}}(1)$ factor.) The amount of extended higher spin supersymmetry in this sense is therefore unconstrained.

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