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All iterated skew polynomial extensions arising from quantized universalenveloping algebras of Kac-Moody algebras are special examples of a very large,axiomatically defined class of algebras, called CGL extensions. For thepurposes of constructing initial clusters for quantum cluster algebrastructures on an algebra R, and classification of the automorphisms of R, oneneeds embeddings of R into quantum tori T which have the property that Rcontains the corresponding quantum affine space algebra A. We explicitlyconstruct such an embedding A \subseteq R \subset T for each CGL extension Rusing the methods of noncommutative noetherian unique factorization domains andrunning a Gelfand-Tsetlin type procedure with normal, instead of centralelements. Along the way we classify the homogeneous prime elements of all CGLextensions and we prove that each CGL extension R has an associated maximaltorus which covers the automorphisms of R corresponding to all normal elements.For symmetric CGL extensions, we describe the relationship between our quantumaffine space algebra A and Cauchon's quantum affine space algebra generated byelements obtained via deleting derivations.

From quantum Ore extensions to quantum tori via noncommutative UFDs