Representations of the affine ageing algebra $\widehat{\mathfrak{a}\mathfrak{g}\mathfrak{e}}(1)$

In this paper, we investigate the affine ageing algebra $\widehat{\mathfrak{a}\mathfrak{g}\mathfrak{e}}(1)$, which is a central extension of the loop algebra of the 1-spatial ageing algebra $\mathfrak{a}\mathfrak{g}\mathfrak{e}(1)$. Certain Verma-type modules including Verma modules and imaginary Verma modules of $\widehat{\mathfrak{a}\mathfrak{g}\mathfrak{e}}(1)$ are studied.Particularly, the simplicity of these modules are characterized and their irreducible quotient modules are determined. We also study the restricted modules of $\widehat{\mathfrak{a}\mathfrak{g}\mathfrak{e}}(1)$ which are also the modules of the affine vertex algebra arising from the 1-spatial ageing algebra $\mathfrak{a}\mathfrak{g}\mathfrak{e}(1)$. We present certain constructions of simple restricted $\widehat{\mathfrak{a}\mathfrak{g}\mathfrak{e}}(1)$-modules and an explicit such example of simple restricted module via the Whittaker module of $\widehat{\mathfrak{a}\mathfrak{g}\mathfrak{e}}(1)$ is given.