Rigidity of quantum tori and the Andruskiewitsch–Dumas conjecture

Abstract

We prove the Andruskiewitsch–Dumas conjecture that the automorphism group of the positive part of the quantized universal enveloping algebra $${\mathcal {U}}_q({\mathfrak {g}})$$ of an arbitrary finite dimensional simple Lie algebra $${\mathfrak {g}}$$ is isomorphic to the semidirect product of the automorphism group of the Dynkin diagram of $${\mathfrak {g}}$$ and a torus of rank equal to the rank of $${\mathfrak {g}}$$ . The key step in our proof is a rigidity theorem for quantum tori. It has a broad range of applications. It allows one to control the (full) automorphism groups of large classes of associative algebras, for instance quantum cluster algebras.