The discriminant criterion and automorphism groups of quantized algebras

Abstract

Abstract. We compute the automorphism groups of some quantized algebras, including tensor products of quantum Weyl algebras and some skew polynomial rings.

Zariski cancellation problem for noncommutative algebras

“…It is easy to check that elements f := x b1 1 · · · x bn n + (cwlt) is effective. We have already seen that there are many examples (those example given in [CPWZ1,CPWZ2]) of noncommutative algebras whose discriminant is dominating and of the form x b1 1 · · · x bn n + (cwlt), whence effective. Here is the main result in this section, which is a slight generalization of Theorem 4.7.…”

“…We recall the definition of the discriminant in the noncommutative setting and everything in this section is taken from [CPWZ1,CPWZ2]. Let R be a commutative algebra and let B and F be algebras both of which contain R as a subalgebra.…”

“…The concept of a dominating element was introduced in [CPWZ1,CPWZ2] to handle the noncommutative AP. We now recall this notion.…”

“…We say a normal element x ∈ B is the greatest common divisor or gcd Some explicit examples of discriminants are given in [CPWZ1,CPWZ2,CYZ1,CYZ2].…”

“…The discriminant method was introduced in [CPWZ1,CPWZ2] to answer the AP for a class of noncommutative algebras. The definition of the discriminant in the noncommutative setting will be reviewed in Section 4.…”