Communications in Mathematical Physics - Volume 231 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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2. The analytic tensor algebra. Let A be a complete bornological algebra, A = + A ⊕ − A the Z -graded algebra of differential forms. The even part + A is a 2 + trivially graded subalgebra. We endow A with a new associative product, the Fedosov product [15], ω1 ω2 = ω1 ω2 − dω1 dω2 , ω1,2 ∈ + A. (6) + A, ) is isomorphic to the nonAssociativity is easy to check. In fact the algebra ( ˆ ⊗n unital tensor algebra T A = n≥1 A , under the correspondence + A a0 da1 . . da2n ←→ a0 ⊗ ω(a1 , a2 ) ⊗ · · · ⊗ ω(a2n−1 , a2n ) ∈ T A, (7) ˆ is the curvature of (ai , aj ).

A path integral approach to the Kontsevich quantization formula. : Free differential algebras: their use in field theory and dual formulation. Lett. Math. Phys. : The cohomology structure of an associative ring. Ann. Math. : Two-dimensional gravity and nonlinear gauge theory. Ann. Phys. : A deformation of three dimensional BF theory. : Deformation quantization of Poisson manifolds, I. : Strongly homotopy Lie algebras. Comm. : Introduction to sh Lie algebras for physicists. Intern’l J. Theor. Phys.

In other words, entire cyclic homology does not distinguish between a complete bornological algebra and its successive nested analytic tensor algebras. This is a particular case of the analytic version [24] of Goodwillie’s theorem [18]. This result is a key point of our bivariant Chern character. The construction of (1) will follow two steps: (a) Using the Goodwillie theorem, we first construct an invertible bivariant class [γ ] ∈ H E0 (A, T A) realizing the equivalence between the entire cyclic homologies of A and T A.

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