From quantum Ore extensions to quantum tori via noncommutative UFDs

TL;DR

This paper constructs explicit embeddings of CGL extensions, arising from quantum groups, into quantum tori, using noncommutative UFD techniques, and classifies their prime elements and automorphisms.

Contribution

It provides a method to embed CGL extensions into quantum tori and classifies their prime elements and automorphisms, advancing understanding of quantum cluster structures.

Findings

Explicit embeddings of CGL extensions into quantum tori are constructed.

Homogeneous prime elements of all CGL extensions are classified.

Each CGL extension has a maximal torus covering all automorphisms.

Abstract

All iterated skew polynomial extensions arising from quantized universal enveloping algebras of Kac-Moody algebras are special examples of a very large, axiomatically defined class of algebras, called CGL extensions. For the purposes of constructing initial clusters for quantum cluster algebra structures on an algebra R, and classification of the automorphisms of R, one needs embeddings of R into quantum tori T which have the property that R contains the corresponding quantum affine space algebra A. We explicitly construct such an embedding A \subseteq R \subset T for each CGL extension R using the methods of noncommutative noetherian unique factorization domains and running a Gelfand-Tsetlin type procedure with normal, instead of central elements. Along the way we classify the homogeneous prime elements of all CGL extensions and we prove that each CGL extension R has an associated maximal torus which covers the automorphisms of R corresponding to all normal elements. For symmetric CGL extensions, we describe the relationship between our quantum affine space algebra A and Cauchon's quantum affine space algebra generated by elements obtained via deleting derivations.