Poincaré duality algebra


Noncommutative geometry

Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



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Generally, a Poincaré duality dg-algebra is a dg-algebra with structure mimicking Poincaré duality in ordinary cohomology.

On the other hand a Poincaré duality C *C^\ast-algebra is a C*-algebra which represents a space in noncommutative topology for which there is a sensible notion of Poincaré duality in K-theory (operator K-theory/K-homology).


For graded-commutative algebras


The structure of a Poincaré duality algebra in dimension nn on a graded-commutative graded associative algebra AA is a linear function ϵ:A nk\epsilon \colon A_n \to k to the ground field such that all the induced bilinear forms

A kA nkA nϵk A_k \otimes A_{n-k} \stackrel{\otimes}{\to} A^n \stackrel{\epsilon}{\to} k

are non-degenerate.

e.g. (Lambrechst-Stanley 07)

For C *C^\ast-algebras

For C*-algebras hence in noncommutative topology there is the following notion of Poincaré duality, which is really Poincaré with respect not to ordinary cohomology but K-theory (operator K-theory).

We start with the definition of Poincaré self-duality and then generalize to Poincaré dual pairs.


A separable C*-algebra AA \in C*Alg is a Poincaré duality algebra (or PD algebra, for short ) if it is dualizable object when regarded as an object of the KK-theory-category, with dual object its opposite algebra.

The element Δ\Delta in def. is called a fundamental class of AA.

This appears as (BMRS 07, def. 2.1, following Connes, p. 601) following (Connes).


Explicitly def. says that AA is a PD algebra if there exists ΔKK(AA op,)\Delta \in KK(A \otimes A^{op}, \mathbb{C}) and Δ KK(,AA op)\Delta^\vee \in KK(\mathbb{C}, A \otimes A^{op}) such that

Δ A opΔ=id AKK(A,A) \Delta^\vee \otimes_{A^{op}} \Delta = id_A \in KK(A,A)


Δ AΔ=id A opKK(A op,A op). \Delta^\vee \otimes_A \Delta = id_{A^{op}} \in KK(A^{op}, A^{op}) \,.

For AA BB two Poincaré duality algebras, def. Cohomological quantization of local prequantum boundary field theory, master thesis, August 2013

Last revised on August 19, 2013 at 16:30:08. See the history of this page for a list of all contributions to it.