## Fusion in von Neumann algebras and loop groups [after A. Wassermann]. (Fusion en algèbres de von Neuman et groupes de lacets [d’après A. Wassermann].)(French)Zbl 0921.46066

Séminaire Bourbaki. Volume 1994/95. Exposés 790-804. Paris: Société Mathématique de France, Astérisque. 237, 251-273, Exp. No. 800 (1996).
In his conference, Vaughan Jones explains the importance of the Connes-Sauvageot tensor product [A. Connes, “Non commutative geometry”, Academic Press (Zbl 0818.46076) Chap. V. App. B, or J.-L. Sauvageot, J. Oper. Theory 9, 237-252 (1983; Zbl 0517.46050)] which is denoted here $$H\otimes_B K$$ where $$B$$ is a von Neumann algebra and $$H$$ (resp. $$K$$) is a right (resp. left) Hilbert $$B$$-module, and which leads to Connes’ fusion. As a constructive example he develops A. J. Wasserman’s work on loop groups [Invent. Math. 133, No. 3, 467-538 (1998) and Proceedings of the International Congress of Mathematicians, ICM 94, Vol. II (1995; Zbl 0854.46055)].
Let $$G$$ be any simply connected compact Lie group, $$I$$ any subinterval of the complex circle $$S^1$$, one defines $$L_IG=\{f:S^1\to G\mid f$$ is $$C^\infty$$ and $$f\equiv 1$$ on $$I^c=S^1\setminus I\}$$, so $$LG=L_{S^1}G$$ is a loop group. For any representation $$(H,\pi)$$ of $$G$$ in a certain class (discrete series) the von Neumann algebra $$\pi(L_IG)''$$ is an hyperfinite factor of type $$\text{III}_1$$ and $$H$$ is a $$\pi(L_I G)''- \pi{(L_{I^c} G)''}^{op}$$ bimodule so one can make the Connes fusion. The case $$G= SU(N)$$ is investigated using vertex operators of V. G. Khnizhnik and A. B. Zamolodchikov [Nucl. Phys. B 247, No. 1, 83-103 (1984; Zbl 0661.17020)]. In particular the inclusion $$\pi(L_I G)''\subset \pi(L_{I^c} G)'$$ gives an example of irreducible inclusion of subfactors of finite and computable index, and for instance for every integer $$\ell$$ one recovers an inclusion of index $$4\cos^2(\pi/ \ell+2)$$.
For the entire collection see [Zbl 0851.00039].

### MSC:

 46L37 Subfactors and their classification 46L35 Classifications of $$C^*$$-algebras 22E67 Loop groups and related constructions, group-theoretic treatment 81T05 Axiomatic quantum field theory; operator algebras 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 22E70 Applications of Lie groups to the sciences; explicit representations 58D05 Groups of diffeomorphisms and homeomorphisms as manifolds

### Citations:

Zbl 0818.46076; Zbl 0517.46050; Zbl 0854.46055; Zbl 0661.17020
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