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

@article{Ip2017ClusterRO,
title={Cluster Realization of Positive Representations of a Split Real Quantum Borel Subalgebra},
author={Ivan C. H. Ip},
journal={Theoretical and Mathematical Physics},
year={2017},
volume={198},
pages={215-238}
}
• I. Ip
• Published 30 November 2017
• Mathematics
• Theoretical and Mathematical Physics
Abstractour previous work, we studied positive representations of split real quantum groups $$\mathcal{U}_{q\widetilde{q}}(\mathfrak{g}_\mathbb{R})$$Uqq~(gR) 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…
2 Citations
• I. Ip
• Mathematics
Selecta Mathematica
• 2018
For each simple Lie algebra $$\mathfrak {g}$$g, we construct an algebra embedding of the quantum group $$\mathcal {U}_q(\mathfrak {g})$$Uq(g) into certain quantum torus algebra $$\mathcal • I. Ip • Materials Science Selecta Mathematica • 2018 For each simple Lie algebra g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} ## References SHOWING 1-10 OF 28 REFERENCES • Mathematics • 2011 We construct a special principal series representation for the modular double U_{q\tilde{q}}(g_R) of type A_r representing the generators by positive essentially self-adjoint operators satisfying • Mathematics • 2017 We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations \mathcal{P}_\lambda of the quantum group We studied the positive representations P_\lambda of split real quantum groups U_{q\tilde{q}}(g_R) restricted to the Borel subalgebra U_{q\tilde{q}}(b_R). We proved that the restriction is We construct the positive principal series representations for \mathcal{U}_q(\mathfrak{g}_\mathbb{R}) where \mathfrak{g} is of simply-laced type, parametrized by \mathbb{R}_{\geq 0}^r where r We give complete detail of the description of the GNS representation of the quantum plane \cA and its dual \hat{\cA} as a von-Neumann algebra. In particular we obtain a rather surprising result • Mathematics • 2012 We derive the quantum Teichm\"uller space, previously constructed by Kashaev and by Fock and Chekhov, from tensor products of a single canonical representation of the modular double of the quantum • I. Ip • Mathematics Selecta Mathematica • 2018 For each simple Lie algebra$$\mathfrak {g}$$g, we construct an algebra embedding of the quantum group$$\mathcal {U}_q(\mathfrak {g})$$Uq(g) into certain quantum torus algebra$$\mathcal
• Mathematics
• 2003
We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of
We give complete detail of the description of the GNS representation of the quantum plane $$\mathcal{A }$$ and its dual $${\widehat{\mathcal{A }}}$$ as a von Neumann algebra. In particular, we obtain
• Mathematics
Inventiones mathematicae
• 2019
We construct an injective algebra homomorphism of the quantum group $$U_q(\mathfrak {sl}_{n+1})$$Uq(sln+1) into a quantum cluster algebra $$\mathbf {L}_n$$Ln associated to the moduli space of framed