46L54 — 1 articles found.

Free Probability and Complex Cobordism

C. R. Math. Rep. Acad. Sci. Canada Vol. 33 (4) 2011, pp. 116–122
Vol.33 (4) 2011
Roland Friedrich; John McKay Details
(Received: 2011-08-13 )
(Received: 2011-08-13 )

Roland Friedrich, BMS, Humboldt University Berlin, Unter den Linden 6, D-10099 Berlin, Germany; e-mail: rolandf@mathematik.hu-berlin.de

John McKay, Dept. Mathematics, Concordia University, Montreal, Canada H3G 1M8; e-mail: mac@mathstat.concordia.ca


We show how Voiculescu’s \(S\)-transform in free probability relates to the space of complex genera in complex cobordism, and endows it with the structure of an infinite Lie group, with the underlying manifold being isomorphic to the group of automorphisms of the formal disc. In addition, this connection gives a link with \(C^*\)-algebras. The group operation introduced can be understood in terms of non-crossing partitions. We then connect both via the Landweber–Novikov algebra with the Faà di Bruno Hopf algebra and the combinatorial structure of rooted trees. Finally, we provide a bridge between free probability, conformal field theory and the KP-hierarchy.

Nous montrons comment la transformée \(S\) de Voiculescu de la théorie des probabilités libres rapporte à l’espace de genres complexes dans la théorie de cobordisme complexe, et lui confère une structure de groupe de Lie infini, avec la variété sous-jacente isomorphe au groupe des automorphismes du disque formel; plus loin, cette connexion permet une liaison avec les \(C^*\)-algèbres. L’opération de groupe introduite est comprise en termes de partitions non-franchissant. L’algèbre de Landweber–Novikov relie ces deux choses à l’algèbre de Hopf de Faà di Bruno et la combinatoire des arbres enracinés. Enfin, nous fournissons une passerelle entre la théorie des probabilités libres, la théorie conforme des champs et l’hiérarchie KP.

Keywords: bordism and cobordism theories, complex cobordism, conformal field theory., formal group laws, free probability and free operator algebras

AMS Subject Classification: Free probability and free operator algebras 46L54

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