Cluster Realization of Positive Representations of a Split Real Quantum Borel Subalgebra

Abstract

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