Fock Space Representation of the Circle Quantum Group

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 $mathcal{F}_{mathbb{R}}$.