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Unipotent and Nakayama automorphisms of quantum nilpotent algebras | | K.R. Goodearl
; M.T. Yakimov
; | | Date: |
1 Nov 2013 | | 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]. | | Source: | arXiv, 1311.0278 | | Services: | Forum | Review | PDF | Favorites | |
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