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 $U_q(\mathfrak{g})$ of an arbitrary finite dimensional simple Lie algebra g is isomorphic to the semidirect product of the automorphism group of the Dynkin diagram of g and a torus of rank equal to the rank of 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.

31 pages, AMS Latex, v.3 contains an application to the isomorphism problem for the algebras U_q^+(g) suggested by L. Scott, minor changes in v.4

References in corpus (5)

• Cluster structures on quantum coordinate rings
• On the Spectra of Quantum Groups
• From quantum Ore extensions to quantum tori via noncommutative UFDs
• On the automorphism groups of q-enveloping algebras of nilpotent Lie algebras
• Automorphisms of quantum matrices

Cited by in corpus (3)

• From quantum Ore extensions to quantum tori via noncommutative UFDs
• The discriminant criterion and automorphism groups of quantized algebras
• The discriminant controls automorphism groups of noncommutative algebras