The discriminant controls automorphism groups of noncommutative algebras

We use the discriminant to determine the automorphism groups of some noncommutative algebras, and we prove that a family of noncommutative algebras has tractable automorphism groups. There is a long history and an extensive study of the automorphism groups of algebras. Determining the full automorphism group of an algebra is generally a no- toriously difficult problem. For example, the automorphism group of t polynomial ring of three variables is not yet understood, and a remarkable result in this direction is given by Shestakov-Umirbaev (SU) which shows the Nagata automorphism is a wild automorphism. Since 1990s, many researchers have been successfully comput- ing the automorphism groups of interesting infinite-dimensional noncommutative algebras, including certain quantum groups, generalized quantum Weyl algebras, skew polynomial rings and many more - see (AlC, AlD, AnD, BJ, GTK, SAV), which is only a partial list. Recently, by using a rigidity theorem for quantum tori, Yakimov has proved the Andruskiewitsch-Dumas conjecture and the Launois- Lenagan conjecture in (Y1, Y2), each of which determines the automorphism group of a family of quantized algebras with parameter q being not a root of unity. A uniform approach to both the Andruskiewitsch-Dumas conjecture and the Launois- Lenagan conjecture is provided in a preprint by Goodearl-Yakimov (GY). These beautiful results, as well as others, motivated us to look into the automorphism groups of noncommutative algebras.