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

By M. Aizenman (Chief Editor)

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For this generalization to work, Ikeda imposes restrictions on the polynomials WAB which amount to making P a Lie algebra, hence his terminology of “non-linear Lie algebra”. In order to obtain an algebraic structure on P analogous to the usual Lie structure required in gauge theory, Ikeda’s bracket is defined on generators of P by [TA , TB ] = WAB ∈ P and extended to all of P via the Leibniz rule: [TA , ] and [ , TB ] are derivations of the commutative algebra P. Ikeda requires that these polynomials satisfy conditions which make P a Poisson algebra.

The construction of the bivariant Chern character proposed in our paper uses simultaneously the (b, B)-complex and Xcomplex approaches. Also, a third complex will be needed as an intermediate step; we call it the completed de Rham–Karoubi complex δ A . 1. Non-commutative differential forms. Let A be a complete bornological algebra. The algebra of non-commutative differential forms over A is the direct sum A = ˆ n A of the n-dimensional subspaces n A = A⊗A ˆ ⊗n for n ≥ 1 and 0 A = A, n≥0 where A = A⊕C is the unitalization of A.

In our analysis of Ikeda’s example, our space is the space of maps Maps( , V ) and the space 0 is the set of ordered pairs φ = (ψ, h), where: (1) ψ is a mapping from a given two-dimensional manifold into the dual V ∗ of the vector space V , and (2) h is a mapping from the same manifold to T ∗ ⊗ V , which in fact is required to be a section of the vector bundle T ∗ ⊗ V −→ . These mappings are denoted µ locally by ψ(x) = ψA (x)T A and h(x) = hA µ (dx ⊗ TA ), where {TA } is a basis of V A ∗ and {T } is the basis of V dual to {TA }.

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