Quantum cluster algebra structures on quantum nilpotent algebras

Abstract

All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always equal the corresponding upper quantum cluster algebras. Previous approaches to these problems for the construction of (quantum) cluster algebra structures on (quantized) coordinate rings arising in Lie theory were done on a case by case basis relying on the combinatorics of each concrete family. The results of the paper have a broad range of applications to these problems, including the construction of quantum cluster algebra structures on quantum unipotent groups and quantum double Bruhat cells (the Berenstein–Zelevinsky conjecture), and treat these problems from a unified perspective. All such applications also establish equality between the constructed quantum cluster algebras and their upper counterparts. The proofs rely on Chatters’ notion of noncommutative unique factorization domains. Toric frames are constructed by considering sequences of homogeneous prime elements of chains of noncommutative UFDs (a generalization of the construction of Gelfand–Tsetlin subalgebras) and mutations are obtained by altering chains of noncommutative UFDs. Along the way, an intricate (and unified) combinatorial model for the homogeneous prime elements in chains of noncommutative UFDs and their alterations is developed. When applied to special families, this recovers the combinatorics of Weyl groups and double Weyl groups previously used in the construction and categorification of cluster algebras. It is expected that this combinatorial model of sequences of homogeneous prime elements will have applications to the unified categorification of quantum nilpotent algebras. Received by the editor 23 December 2013. 2010 Mathematics Subject Classification. Primary 16T20; Secondary 13F60, 17B37, 14M15.