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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 ascontinuous generalizations of the quantum groups of the Kac-Moody Lie algebrasof finite, respectively affine type $A$. In the present paper, we define theFock space representation $\mathcal{F}_{\mathbb{R}}$ of the quantum group$\mathbf{U}_\upsilon(\mathfrak{sl}(\mathbb{R}))$ as the vector space generatedby real pyramids (a continuous generalization of the notion of partition). Inaddition, 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$\mathcal{F}_{\mathbb{R}}$.

Fock space representation of the circle quantum group