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Somewhat surprisingly, it follows from work of Woronowicz-Zakrewski that the ax + b semigroup over the real numbers admits a quantisation as a locally compact quantum group (as opposed to just a quantum semigroup). In this talk I will review this construction, describe how it leads to a “positive” quantization of SL(2,\mathbb{R}) as a locally compact quantum group, and discuss the connection with positive representations of quantized universal enveloping algebras. This illustrates the simplest special case a more general construction involving the theory of quantum cluster algebras. (Joint work with K. De Commer, G. Schrader and A. Shapiro)