Homotopy of product systems, K-theory of k-graph algebras, and the Yang-Baxter equation

Aidan Sims – 24 February 2021

Higher-rank graphs, or k-graphs, are higher-dimensional analogues of directed graphs. Each k-graph can be described in terms of a coloured graph, called is skeleton, and some factorisation rules that describe how 2-coloured paths pair up into commuting squares. C*-algebras of k-graphs generalise Cuntz–Krieger algebras, and have been the subject of sustained interest essentially because questions about crossed products of C*-algebras by higher-rank free abelian groups are hard, and k-graphs algberas constitute a comparably tractable class of examples that could point the way to general theorems. A particularly obstinate question in this vein is that of determining the K-theory of a k-graph algebra; or even just whether the K-theory depends on the factorisation rules, or only on the skeleton. I’ll outline some joint work with James Fletcher and Elizabeth Gillaspy that uses a homotopy argument to establish a surprising link between this question and the question of connectedness (or otherwise) of the space of solutions to a Yang-Baxter-like equation. I won’t assume any background about C*-algebras, k-graphs, or the Yang-Baxter equations, and all are welcome. People who might know about connectedness (or otherwise) of the spaces of solutions to Yang-Baxter-like equations are especially welcome!