2013/11/01 by K. R. Goodearl, Goodearl, K. R., Milen Yakimov +1 · 1 citation
Mathematics · #16W20 #17B37 #20G42 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Primary 16T20 #Quantum Algebra (math.QA) #Rings and Algebras (math.RA) #Secondary 16S36
paper · pdf · doi:10.48550/arxiv.1311.0278
openalex publication_date 2013/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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].