Q and T systems, paths and cluster positivity II

We present non-commutative generalizations of discrete integrable systems that possess the positive Laurent property. This is motivated by a Laurent positivity conjecture of M. Kontsevitch, regarding a non-commutative version of rank two cluster algebras, for which we give a proof in the affine cases. Our solution uses a non-commutative generalization of the path models used to solve classical Q and T systems. More generally, we will show how to use the theory of quasi-determinants of Gelfand et al to define non-commutative mutations within the path model framework, as local rearrangements of non-commutative branching continued fractions. This is joint work with Rinat Kedem.