On noncommutative finite factorization domains

Bell, Jason; Heinle, Albert; Levandovskyy, Viktor

doi:10.1090/tran/6727

On noncommutative finite factorization domains
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by Jason P. Bell, Albert Heinle and Viktor Levandovskyy PDF

Trans. Amer. Math. Soc. 369 (2017), 2675-2695 Request permission

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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