Geodesic
Cluster realization of positive representations of a~split real quantum Borel subalgebra
Teoretičeskaâ i matematičeskaâ fizika, Tome 198 (2019) no. 2, pp. 246-272

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In our previous work, we studied positive representations of split real quantum groups $\mathcal U_{q\tilde q}(\mathfrak g_{\mathbb R})$ 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.
Keywords: positive representation, split real quantum group, modular double, quantum cluster algebra, tensor category. 
@article{TMF_2019_198_2_a3,
     author = {I. Ch.-H. Ip},
     title = {Cluster realization of positive representations of a~split real quantum {Borel} subalgebra},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {246--272},
     publisher = {mathdoc},
     volume = {198},
     number = {2},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2019_198_2_a3/}
}
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I. Ch.-H. Ip. Cluster realization of positive representations of a~split real quantum Borel subalgebra. Teoretičeskaâ i matematičeskaâ fizika, Tome 198 (2019) no. 2, pp. 246-272. http://geodesic.mathdoc.fr/item/TMF_2019_198_2_a3/