Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum $SL^+(N,\mathbb{R})$ as a locally compact quantum group

K. De Commer, G. Schrader, A. Shapiro, C. Voigt

TL;DR

The paper constructs a higher-rank, purely continuous, $q$-deformed locally compact quantum group, $SL^+_\hbar(N,\mathbb{R})$, by unimodular Drinfeld–Jimbo quantization of the totally positive part of $SL^+(N,\mathbb{R})$. It develops a quantum cluster framework based on Fock–Goncharov flips, Gaussians, and the quantum dilogarithm to produce a modular multiplicative unitary underlying the quantum group structure, Haar weights, and modular data. The von Neumann algebras are realized as twisted group von Neumann algebras, and explicit formulas for modular element, antipode, scaling, and duality are derived, with the Drinfeld double yielding the full quantized Borel and resulting $SL^+_\hbar(N,\mathbb{R})$. This analytic foundation advances the category of positive representations and links to higher Teichmüller theory and potential topological quantum field theory structures. Overall, the work provides a concrete pathway from quantum cluster realizations to globally defined analytic quantum groups in higher rank audiences.

Abstract

We construct the first examples of purely continuous, $q$-deformed Lie type locally compact quantum groups in higher rank. They arise from Drinfeld-Jimbo quantization, at unimodular deformation parameter, of the totally positive part of higher rank split real Lie groups in type $A$. Our techniques are based on quantum cluster theory, in particular as developed through the work of Fock and Goncharov.

Quantum $SL^+(N,\mathbb{R})$ as a locally compact quantum group

TL;DR

The paper constructs a higher-rank, purely continuous, -deformed locally compact quantum group, , by unimodular Drinfeld–Jimbo quantization of the totally positive part of . It develops a quantum cluster framework based on Fock–Goncharov flips, Gaussians, and the quantum dilogarithm to produce a modular multiplicative unitary underlying the quantum group structure, Haar weights, and modular data. The von Neumann algebras are realized as twisted group von Neumann algebras, and explicit formulas for modular element, antipode, scaling, and duality are derived, with the Drinfeld double yielding the full quantized Borel and resulting . This analytic foundation advances the category of positive representations and links to higher Teichmüller theory and potential topological quantum field theory structures. Overall, the work provides a concrete pathway from quantum cluster realizations to globally defined analytic quantum groups in higher rank audiences.

Abstract

We construct the first examples of purely continuous, -deformed Lie type locally compact quantum groups in higher rank. They arise from Drinfeld-Jimbo quantization, at unimodular deformation parameter, of the totally positive part of higher rank split real Lie groups in type . Our techniques are based on quantum cluster theory, in particular as developed through the work of Fock and Goncharov.
Paper Structure (9 sections, 14 theorems, 128 equations, 17 figures)

This paper contains 9 sections, 14 theorems, 128 equations, 17 figures.

Table of Contents

  1. Skew-symmetric spaces and their representations
  2. Skew-symmetric spaces
  3. Representations of skew-symmetric spaces
  4. Gaussians, quantum dilogarithms and quantum exponential maps
  5. Pentagonal Fock--Goncharov flip
  6. Skew-symmetric of type An
  7. Colored braid graphs
  8. Partition functions and standard generators via form sums
  9. R-matrix and the braided Fock--Goncharov flip

Key Result

Theorem 1.7

If $V$ is symplectic, all its Heisenberg $\hbar$-representations are irreducible and unitarily equivalent. Any other unitary $\hbar$-representation is a (possibly infinite) direct sum of Heisenberg $\hbar$-representations.

Figures (17)

  • Figure 1: The diagram $C_4$ with the diagram $C_4'$ inside.
  • Figure 2: The vectors $\overset{\scaleobj{0.7}{\nearrow}}{e}_{3,2}$, $\overset{\scaleobj{0.7}{\searrow}}{e}_{1,1}$ and $\overset{\scaleobj{0.7}{\swarrow}}{e}_{1,0}$ as (undirected) paths in $C_4$.
  • Figure 3: The diagram $\overset{\scaleobj{0.7}{\nearrow}}{C}_4$ inside $C_4$.
  • Figure 4: The diagram $\overset{\scaleobj{0.7}{\searrow}}{C}\,\!'_4$ inside $C_4$.
  • Figure 5: Local graph modification for braid move $\sigma_{i+1}\sigma_{i}\sigma_{i+1}\mapsto \sigma_{i}\sigma_{i+1}\sigma_{i}$.
  • ...and 12 more figures

Theorems & Definitions (48)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Example 1.5
  • Example 1.6
  • Theorem 1.7: Stone-von Neumann theorem
  • Definition 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 38 more