Fock space representation of the circle quantum group

In [arXiv:1711.07391] we have defined quantum groups ${\mathbf{U}}_{\upsilon }(\mathfrak{s}\mathfrak{l}(\mathbb{R}))$ and ${\mathbf{U}}_{\upsilon }(\mathfrak{s}\mathfrak{l}(S^1))$, 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 ${\mathbf{U}}_{\upsilon }(\mathfrak{s}\mathfrak{l}(\mathbb{R}))$ 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 ${\mathbf{U}}_{\upsilon }(\mathfrak{s}\mathfrak{l}(S^1))$ on ${\mathcal{F}}_{\mathbb{R}}$.