Duality for finite bipartite graphs (with an application to II₁ factors)

Marie Choda

Vol. 158 (1993), No. 1, 49–65

DOI: 10.2140/pjm.1993.158.49

Abstract

Let Γ be a finite graph with bicolored vertices and 𝜃 a color-preserving automorphism of Γ. We define the dual graph Γ = Γ(𝜃) of Γ by 𝜃 and the dual 𝜃 of 𝜃 which is an automorphism of Γ. Under some conditions, Γ is isomorphic to Γ. A bicolored graph gives two weighted graphs. The following pair of graphs treated in Index theory are dual pairs: {Coxeter graph of type A_2n−3, D_n }, {A_2n−5⁽¹⁾,D_n⁽¹⁾}, {E₆⁽¹⁾,E₇⁽¹⁾}, {D_l⁽¹⁾,D_2l−2⁽¹⁾}, and {4-star S(1,1,k + 1,k + 1), Γ_k}. The graph of type D₄ or E₆ is self dual, but as a weighted graph, the dual of it is another one.

As applications, we have two kinds of outer automorphisms with the period 2 on inclusions of hyperfinite II₁ factors, one of which gives the inclusion of the crossed products isomorphic to the original one and the other gives the inclusion not isomorphic to the original one.

Mathematical Subject Classification 2000

Primary: 46L37

Secondary: 05C25, 46L35

Milestones

Received: 20 October 1990

Revised: 15 March 1992

Published: 1 March 1993