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Transactions of the American Mathematical Society

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On noncommutative finite factorization domains


Authors: Jason P. Bell, Albert Heinle and Viktor Levandovskyy
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 16U10, 16U30, 16W70
Published electronically: August 3, 2016
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Abstract: A domain $ R$ is said to have the finite factorization property if every nonzero nonunit element of $ R$ has at least one and at most finitely many distinct factorizations up to multiplication of irreducible factors by central units. Let $ k$ be an algebraically closed field and let $ A$ be a $ k$-algebra. We show that if $ A$ has an associated graded ring that is a domain with the property that the dimension of each homogeneous component is finite, then $ A$ is a finite factorization domain. As a corollary, we show that many classes of algebras have the finite factorization property, including Weyl algebras, enveloping algebras of finite-dimensional Lie algebras, quantum affine spaces and shift algebras. This provides a termination criterion for factorization procedures over these algebras. In addition, we give explicit upper bounds on the number of distinct factorizations of an element in terms of data from the filtration.


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Additional Information

Jason P. Bell
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Email: jpbell@uwaterloo.ca

Albert Heinle
Affiliation: David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Email: aheinle@uwaterloo.ca

Viktor Levandovskyy
Affiliation: Lehrstuhl D für Mathematik, RWTH Aachen University, 52062 Aachen, Germany
Email: viktor.levandovskyy@math.rwth-aachen.de

DOI: https://doi.org/10.1090/tran/6727
Keywords: Finite factorization domains, Weyl algebras, quantum affine spaces, Lie algebras, factorization
Received by editor(s): November 4, 2014
Received by editor(s) in revised form: April 9, 2015, and April 16, 2015
Published electronically: August 3, 2016
Additional Notes: The first two authors thank NSERC for its generous support
Article copyright: © Copyright 2016 American Mathematical Society