  In [arXiv:1711.07391] we have defined quantum groups
$\mathbf{U}_\upsilon(\mathfrak{sl}(\mathbb{R}))$ and
$\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$, which can be interpreted as
continuous generalizations of the quantum groups of the Kac-Moody Lie algebras
of finite, respectively affine type $A$. In the present paper, we define the
Fock space representation $\mathcal{F}_{\mathbb{R}}$ of the quantum group
$\mathbf{U}_\upsilon(\mathfrak{sl}(\mathbb{R}))$ as the vector space generated
by real pyramids (a continuous generalization of the notion of partition). In
addition, by using a variant of the "folding procedure" of Hayashi-Misra-Miwa,
we define an action of $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$ on
$\mathcal{F}_{\mathbb{R}}$.
