A noncommutative theory of Penrose tilings. (English) Zbl 1087.52509
In 1986, the first author introduced quantales as an invariant of noncommutative \(C^\ast\)-algebras – they should be thought as noncommutative topological spaces. This paper uses quantales to study Penrose tilings. Since quantales are algebraic structures, they can be defined by generators and relations. It is possible to associate a quantale to Penrose tilings; classically one gets a topological space. Algebraically irreducible representations of this quantale then classify Penrose tilings. This approach is analogous to that of Connes’ noncommutative teometry who uses \(C^\ast\)-algebras.
Reviewer: Jiří Rosický (Brno)
MSC:
| 52C23 | Quasicrystals and aperiodic tilings in discrete geometry |
| 58B34 | Noncommutative geometry (à la Connes) |
| 46L85 | Noncommutative topology |
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