Nonrelativistic spin-3 symmetries in 2+1 dimensions from expanded and extended Nappi-Witten algebras

Abstract

We show that infinite families of Galilean spin-3 symmetries in $2+1$ dimensions, which include higher-spin extensions of the Bargmann, Newton-Hooke, nonrelativistic Maxwell, and nonrelativistic AdS-Lorentz algebras, can be obtained as Lie algebra expansions of two different spin-3 extensions of the Nappi-Witten symmetry. These higher-spin Nappi-Witten algebras, in turn, are obtained by means of In\"on\"u-Wigner contractions applied to suitable direct product extensions of $\mathfrak{s}\mathfrak{l}(3,\mathbb{R})$. Conversely, we show that the same result can be obtained by considering contractions of expanded $\mathfrak{s}\mathfrak{l}(3,\mathbb{R})$ algebras. The method can be used to define nonrelativistic higher-spin Chern-Simon gravity theories in $2+1$ dimensions in a systematic way.