The q-Division Ring: Fixed Rings and Automorphisms.

Sian Fryer (University of Manchester)

Wednesday 25th June, 2014 16:00-17:00 Maths 204


The division ring of the q-commuting polynomials in two variables is one of the easiest examples of noncommutative infinite dimensional division rings to define, but answering even fairly basic questions concerning the structure of its automorphism group or its sub-division rings of finite index is still quite difficult.  The second question in particular is of interest due to its connections with Artin's conjectured classification of surfaces in non-commutative algebraic geometry. 

I will describe the structure of the fixed rings of the q-division ring for a certain class of finite groups and use this to construct some rather unexpected examples of homomorphisms on it, including a conjugation automorphism which is not inner and a conjugation homomorphism which is not even bijective.  If there's time I'll also talk about how we can view the q-division ring as a deformation of a commutative field with a Poisson bracket, and ways to use this as an alternative approach to understanding the fixed rings.

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