Title: The prime spectrum of quantum matrices

 

Abstract: We start with a very classical question in ring/representation theory: given an algebra, classify the prime ideals and describe the Zariski topology of the prime spectrum. For many nice noncommutative algebras, and in particular the algebras of quantum matrices (that is, the standard quantization of the coordinate rings of matrices), we can tackle this problem by first understanding finitely many graded-prime (or ``H-prime'') ideals in the algebra. The H-primes stratify the prime spectrum, reducing the question to two smaller problems: describe the individual strata (essentially solved), and understand how to reassemble these pieces back into a coherent whole (still very much open). In particular, this approach turns a topology-flavoured question into a collection of purely algebraic ones: mostly computing quotients, localizations, and centres of various algebras.

 

This will be a talk in two parts: in the first week I'll talk about the background and general theory of H-stratification, and then in the second week go on to talk about my own work in tackling some of the questions described above for certain nice cases. The first week in particular should be accessible for non-experts and grad students.