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Scalar Fields in 3D Asymptotically Flat Higher-Spin Gravity

In this work we construct a novel associative algebra and use it to define a theory of higher-spin gravity in (2+1)-dimensional asymptotically flat spacetimes. Our construction is based on a quotient of the universal enveloping algebra (UEA) of $\mathfrak{isl}(2,\mathbb{R})$ with respect to the ideal generated by its Casimir elements, the mass squared $\mathcal{M}^2$ and the three-dimensional analogue of the square of the Pauli-Lubanski vector $\mathcal{S}$ and propose to call the resulting associative algebra $\mathfrak{ihs}(\mathcal{M}^2,\mathcal{S})$. We provide a definition of its generators and even though we are not yet able to provide the complete set of multiplication rules of this algebra our analysis allows us to study many interesting and relevant sub-structures of $\mathfrak{ihs}(\mathcal{M}^2,\mathcal{S})$. We then show how to consistently couple a scalar field to an $\mathfrak{ihs}(\mathcal{M}^2,\mathcal{S})$ higher-spin gauge theory.