On Noncommutative Finite Factorization Domains

TL;DR

This paper establishes conditions under which noncommutative algebras like Weyl and quantum algebras have finite factorizations, providing a termination criterion and explicit bounds for factorization procedures.

Contribution

It proves that certain noncommutative algebras are finite factorization domains based on their graded structures, extending known classes.

Findings

Many important algebras have the finite factorization property.

Provides explicit upper bounds on the number of factorizations.

Offers a termination criterion for factorization algorithms.

Abstract

A domain $R$ is said to have the finite factorization property if every nonzero non-unit 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.