Cluster realization of positive representations of split real quantum Borel subalgebra

TL;DR

This paper explicitly constructs the tensor product decomposition of positive representations of split real quantum Borel subalgebras using cluster mutations, providing a concrete realization of the abstract decomposition previously known.

Contribution

It introduces a cluster realization approach to explicitly decompose tensor products of positive quantum group representations, advancing understanding of their structure.

Findings

Explicit tensor product decomposition via cluster mutations

Realization of the decomposition as quiver mutations

Enhanced understanding of positive representations structure

Abstract

In our previous work, we studied the positive representations of split real quantum groups $\mathcal{U}_{q\tilde{q}}(\mathfrak{g}_\mathbb{R})$ restricted to its Borel part, and showed that they are closed under taking tensor products. However, the tensor product decomposition was only constructed abstractly using the GNS-representation of a $C^*$-algebraic version of the Drinfeld-Jimbo quantum groups. In this paper, using the recently discovered cluster realization of quantum groups, we write down the decomposition explicitly by realizing it as a sequence of cluster mutations in the corresponding quiver diagram representing the tensor product.