Unipotent and Nakayama automorphisms of quantum nilpotent algebras

Abstract

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 [20, 22] 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 [13, 20].

24 pages

References in corpus (6)

• Calabi-Yau algebras
• On the Spectra of Quantum Groups
• The Launois-Lenagan conjecture
• From quantum Ore extensions to quantum tori via noncommutative UFDs
• Twisted Calabi-Yau property of Ore extensions
• The discriminant controls automorphism groups of noncommutative algebras

Cited by in corpus (6)

• The discriminant criterion and automorphism groups of quantized algebras
• Discriminants and Automorphism Groups of Veronese subrings of skew polynomial rings
• Nakayama automorphism and applications
• Automorphism groupoids in noncommutative projective geometry
• The discriminant controls automorphism groups of noncommutative algebras
• Automorphisms for Some "symmetric" Multiparameter Quantized Weyl Algebras and Their Localizations