[Paper Review] On tensor products of positive representations of split real quantum Borel subalgebra $U_{q ilde{q}}(b_R)$

Ivan Chi-Ho Ip|arXiv (Cornell University)|May 19, 2014

Algebraic structures and combinatorial models25 references3 citations

TL;DR

This paper establishes that the positive representations of the split real quantum Borel subalgebra $υ_{q\tilde{q}}(\mathfrak{b}_{\mathbb{R}})$ are independent of the parameter $\lambda$, and proves that their tensor product decomposes via a GNS-representation of the multiplier Hopf algebra, using the multiplicative unitary. The key result is a unitary equivalence of the tensor product to $L^2(\mathbb{R}^{n(n+1)/2}) \otimes \mathcal{P}$, with explicit intertwiners constructed from quantum dilogarithm functions.

ABSTRACT

We studied the positive representations $P_λ$ of split real quantum groups $U_{q ilde{q}}(g_R)$ restricted to the Borel subalgebra $U_{q ilde{q}}(b_R)$. We proved that the restriction is independent of the parameter $λ$. Furthermore, we prove that it can be constructed from the GNS-representation of the multiplier Hopf algebra $U_{q ilde{q}}^{C^*}(b_R)$ constructed earlier, which enables us to decompose their tensor product using the theory of the "multiplicative unitary". This will be an essential ingredient in the construction of quantum higher Teichmüller theory from the perspective of representation theory, generalizing earlier work by Frenkel-Kim.

Motivation & Objective

• To understand the structure of tensor products of positive representations of split real quantum groups restricted to the Borel subalgebra.
• To resolve the open problem of tensor product decomposition in higher-rank split real quantum groups.
• To generalize the quantum dilogarithm-based fusion rules from $\mathfrak{sl}(2,\mathbb{R})$ to higher-rank Lie algebras like $\mathfrak{sl}_3$.
• To construct explicit intertwiners using the multiplicative unitary and $g_b$-functions for the Borel part of the quantum group.
• To provide a representation-theoretic foundation for quantum higher Teichmüller theory via the modular double and positive representations.

Proposed method

• Restrict the positive representations $\mathcal{P}_\lambda$ of $\mathcal{U}_{q\tilde{q}}(\mathfrak{g}_{\mathbb{R}})$ to the Borel subalgebra $\mathcal{U}_{q\tilde{q}}(\mathfrak{b}_{\mathbb{R}})$, proving independence from $\lambda$.
• Use the GNS-representation of the multiplier Hopf algebra $\mathcal{U}_{q\tilde{q}}^{C^*}(\mathfrak{b}_{\mathbb{R}})$ to realize the representations on a Hilbert space.
• Construct explicit unitary transformations $\Phi_2$ using products of quantum dilogarithm operators $g_b$ to relate the tensor product to a standard form.
• Apply the multiplicative unitary formalism to decompose the tensor product representation into a direct integral over $L^2(\mathbb{R}^{n(n+1)/2}) \otimes \mathcal{P}$.
• Verify that the transformed action matches $1 \otimes \mathbf{e}_i$ and $K_i \otimes K_i$, confirming the decomposition.

Experimental results

Research questions

• RQ1 Is the restriction of positive representations $\mathcal{P}_\lambda$ to the Borel subalgebra $\mathcal{U}_{q\tilde{q}}(\mathfrak{b}_{\mathbb{R}})$ independent of the parameter $\lambda$?
• RQ2 Can the tensor product of two positive representations of $\mathcal{U}_{q\tilde{q}}(\mathfrak{b}_{\mathbb{R}})$ be decomposed using the GNS-representation and multiplicative unitary?
• RQ3 What is the explicit form of the intertwiners that realize the tensor product decomposition for $\mathfrak{sl}_3$?
• RQ4 How does the decomposition generalize to higher-rank Lie algebras beyond $A_n$?
• RQ5 Can the quantum mutation operator in higher Teichmüller theory be constructed from the multiplicity module of this tensor product?

Key findings

• The restriction of positive representations $\mathcal{P}_\lambda$ to the Borel subalgebra $\mathcal{U}_{q\tilde{q}}(\mathfrak{b}_{\mathbb{R}})$ is independent of $\lambda$, allowing a uniform description.
• The tensor product $\mathcal{P}_{\lambda_1} \otimes \mathcal{P}_{\lambda_2}$ is unitarily equivalent to $L^2(\mathbb{R}^{n(n+1)/2}) \otimes \mathcal{P}$ for type $A_n$, with explicit intertwiners constructed via $g_b$-functions.
• For $\mathfrak{sl}_3$, the transformation $\Phi_2$ explicitly realizes the equivalence $\Delta(\mathbf{e}_i) \simeq 1 \otimes \mathbf{e}_i$ and $\Delta(K_i) \simeq K_i \otimes K_i$.
• The intertwiners are expressed as products of quantum dilogarithm operators $g_b$, with arguments that are positive and $q$-commuting, ensuring well-definedness.
• The construction generalizes the $\mathfrak{sl}(2,\mathbb{R})$ case, where the Plancherel measure is given by the quantum dilogarithm, to higher-rank algebras.
• The procedure suggests a connection to the Heisenberg double, and provides a representation-theoretic framework for quantum higher Teichmüller theory.