After Garoufalidis–Lê–Zeilberger's quantum extension (2006), a number of noncommutative extensions were developed by Foata–Han, Konvalinka–Pak, and Etingof–Pak.
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Its applications deal with domains where privileged directions appear, like concurrent processes, traffic networks, spacetime models, noncommutative geometry, rewriting systems and the modelling of biological systems.
Many topological and geometric properties of the classical 2-torus have algebraic analogues for the noncommutative tori, and as such they are fundamental examples of a noncommutative space in the sense of Alain Connes.
Noncommutative geometry, quantum fields and motives. (with Alain Connes), American Mathematical Society, Colloquium Publications, 2008.