Higher Rank Analytic Toeplitz Algebras

To a higher rank directed graph (a so called k-graph) in the sense of Kumjian and Pask, one can associate natural noncommutative analytic Toeplitz algebras, both weakly closed and norm closed. I shall describe some simple (finitely generated) 2-graphs (they have red and blue edges) and methods for the classification of their algebras. In the case of single vertex graphs the algebras are generated by 2 families of isometries (on a Fock space) subject to red-blue commutations relations. A useful intermediate invariant is the character (Gelfand) space, which we explicitly realise as an algebraic variety subset of a product of two unit balls.