Unipotent and Nakayama automorphisms of quantum nilpotent algebras

Summary

Automorphisms of algebras R from a very large axiomatic class of quantum nilpotent algebras are studied using techniques from noncommutative unique factorization domains and quantum cluster algebras. First, the Nakayama automorphism of R (associated to its structure as a twisted Calabi–Yau algebra) is determined and shown to be given by conjugation by a normal element, namely, the product of the homogeneous prime elements of R (there are finitely many up to associates). Second, in the case when R is connected graded, the unipotent automorphisms of R are classified up to minor exceptions. This theorem is a far reaching extension of the classification results previously used to settle the Andruskiewitsch–Dumas and Launois–Lenagan conjectures. The result on unipotent automorphisms has a wide range of applications to the determination of the full automorphisms groups of the connected graded algebras in the family. This is illustrated by a uniform treatment of the automorphism groups of the generic algebras of quantum matrices of both rectangular and square shape.

This paper is devoted to a study of automorphisms of quantum nilpotent algebras, a large, axiomatically defined class of algebras. The algebras in this class are known under the name Cauchon–Goodearl–Letzter extensions and consist of iterated skew polynomial rings satisfying certain common properties for algebras appearing in the area of quantum groups. The class contains the quantized coordinate rings of the Schubert cells for all simple algebraic groups, multiparameter quantized coordinate rings of many algebraic varieties, quantized Weyl algebras, And related algebras. The quantized coordinate rings of all double Bruhat cells are localizations of special algebras in the class.