Quantized groups, string algebras and Galois theory for algebras. (English) Zbl 0696.46048

Ocneanu, Adrian

Operator algebras and applications. Vol 2: Mathematical physics and subfactors, Pap. UK-US Jt. Semin., Warwick/UK 1987, Lond. Math. Soc. Lect. Note Ser. 136, 119-172 (1988).

[For the entire collection see Zbl 0668.00015.]
Introducing the notion of a paragroup (a natural quantization of a finite group) the author builds a Galois theory for the relative position of subfactors of finite index in the hyperfinite \(II_ 1\)-factor R. These paragroups whose underlying set is a graph with elements the strings on the graph and composition given by geometrical connection produce a complete conjugacy invariant. This construction also explains and proves the rigidity of the James index: for \(n\geq 3\) there are at most 4 conjugacy classes of subfactors of R with James index 4 \(cos^ 2(\pi /n)\).

Reviewer: H.Schröder

MSC:

46L35	Classifications of \(C^*\)-algebras
46L60	Applications of selfadjoint operator algebras to physics
81T30	String and superstring theories; other extended objects (e.g., branes) in quantum field theory

Keywords:

paragroup; Galois theory for the relative position of subfactors of finite index in the hyperfinite \(II_ 1\)-factor; rigidity of the James index

Citations:

Zbl 0668.00015