\contentsline {section}{\numberline {1}Introduction}{1}
\contentsline {section}{\numberline {2}Cluster ensembles}{3}
\contentsline {subsection}{\numberline {2.1}Basic definitions}{3}
\contentsline {subsection}{\numberline {2.2}Connections between quantum ${\cal X}$-varieties}{4}
\contentsline {section}{\numberline {3}Motivations: the $\ast $-quantization of cluster ${\cal X}$-varieties}{6}
\contentsline {subsection}{\numberline {3.1}$\ast $-quantization of the space ${\cal X}^+$ via the quantum logarithm and}{6}
\contentsline {subsection}{\numberline {3.2}Modular double of ${\cal X} $ and quantization of ${\cal X}^+$}{8}
\contentsline {section}{\numberline {4}The intertwiner}{10}
\contentsline {subsection}{\numberline {4.1}A bimodule structure on functions on the ${\cal A}$--space}{10}
\contentsline {subsection}{\numberline {4.2}The intertwiner via the quantum dilogarithm }{11}
\contentsline {section}{\numberline {5}The quantum logarithm and dilogarithm functions}{14}