Communications In Mathematical Physics - Volume 300 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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Theorem 1 ([10], [4]). The following identities hold true: ch(x, y) = x + y + w n≥1 (5) simple geometric Lie type (n,2) duf(x, y) = n≥1 (x, y), simple geometric wheel type (n,2) w m (x, y), (6) where m is the order of the symmetry group of the graph . Here geometric means that graphs are not labeled. Note that the definition of both (x, y) and w requires an order on the set of edges, but the product w (x, y) is independent of this order. Even though ch(x, y) and duf(x, y) are defined over rationals, some of the coefficients w are very probably irrational (see example of [8]).

In more detail, T Autn consists of automorphisms of lien such that xi → Ad gi xi = gi xi gi−1 , where gi ∈ exp(lien ). Similarly, elements of n S Autn are tangential automorphisms of lien with an extra property x = i=1 xi → x. The family of Lie algebras tdern is equipped with simplicial Lie homomorphisms tdern → tdern+1 . For instance, for u = (a, b) ∈ tder2 we define u 1,2 = (a(x, y), b(x, y), 0), u 2,3 = (0, a(y, z), b(y, z)), u 12,3 = (a(x + y, z), a(x + y, z), b(x + y, z)), and similarly for other simplicial maps.

The ξ -deformation of the Campbell-Hausdorff series chξ (x, y) is defined as chξ (x, y) = x + y + w (ξ ) (x, y). n≥1 (7) simple geometric Lie type (n,2) In a similar fashion, we introduce a deformation of the Duflo function, duf ξ (x, y) = n≥1 simple geometric wheel type (n,2) w (ξ ) (x, y). m For ξ = (0, 1) (the right corner of the eye on Fig. 1), the expression (7) is given by the standard Campbell-Hausdorff series, and for ξ in the position α on the iris, the Kontsevich Vanishing Lemma implies chα (x, y) = x +y.

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