Fock space representation of the circle quantum group

TL;DR

This paper introduces a Fock space representation for a continuous quantum group related to the circle, extending classical quantum group concepts to a continuous setting and defining their actions on this new space.

Contribution

It constructs a Fock space for the circle quantum group and defines its action, generalizing discrete quantum group representations to a continuous framework.

Findings

Defined the Fock space $\

_{}$$ for the circle quantum group.

Established an action of $\

Abstract

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