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We prove the Andruskiewitsch-Dumas conjecture that the automorphism group ofthe positive part of the quantized universal enveloping algebra$U_q({\mathfrak{g}})$ of an arbitrary finite dimensional simple Lie algebra gis isomorphic to the semidirect product of the automorphism group of the Dynkindiagram of g and a torus of rank equal to the rank of g. The key step in ourproof is a rigidity theorem for quantum tori. It has a broad range ofapplications. It allows one to control the (full) automorphism groups of largeclasses of associative algebras, for instance quantum cluster algebras.

Rigidity of quantum tori and the Andruskiewitsch-Dumas conjecture