Involutive Rings and Algebras

Commutative and non-commutative rings with an anti involution have been studied in many different contexts. In this talk I will give you a brief overview of some of the results and constructions from my recent and ongoing work joint with Zachary Greenberg and Anna Wienhard studying the algebraic properties of symplectic groups defined over involutive rings, including notions of traces, Caley Hamilton identities, Hopf algebra structures and some ways to realise these constructions within non-commutative cluster algebras.

If time, I will also mention some new ideas about mutation of matrices with entries in an involutive algebra.