Seminar

CGL (Cauchon-Goodearl-Letzter) extensions form a large, axiomatically defined class of iterated Ore extensions.
Various important families, such as quantum Schubert cell algebras and quantum Weyl algebras, arise as special cases.
From the point of view of noncommutative algebra, this is the "best" current definition of quantum nilpotent algebras. We will describe a general contruction of quantum clusters on all of these algebras via noncommutative unique factorization domains. The structure of their prime elements sees combinatorics that was previously specific to Weyl groups.
Another aspect of this work is that quantum clusters are intrinsically constructed as unique families of prime elements of chains of subalgebras, while previous constructions relied on direct arguments with quantum minors.
This is a joint work with Ken Goodearl (UC Santa Barbara).