Quantum SL⁺(N,ℝ) as a locally compact quantum group

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.