We present all the symmetry superalgebras $mathfrak{g}$ of all warpedAdS$_kimes_w M^{d-k}$, $k>2$, flux backgrounds in $d=10, 11$ dimensionspreserving any number of supersymmetries. First we give the conditions for$mathfrak{g}$ to decompose into a direct sum of the isometry algebra ofAdS$_k$ and that of the internal space $M^{d-k}$. Assuming this decomposition,we identify all symmetry superalgebras of AdS$_3$ backgrounds by showing thatthe isometry groups of internal spaces act transitively on spheres. Wedemonstrate that in type II and $d=11$ theories the AdS$_3$ symmetrysuperalgebras may not be simple and also present all symmetry superalgebras ofheterotic AdS$_3$ backgrounds. Furthermore, we explicitly give the symmetrysuperalgebras of AdS$_k$, $k>3$, backgrounds and prove that they are allclassical.
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