nLab
Poincaré duality algebra

Context

Noncommutative geometry

Operator algebra

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

Introduction

Concepts

field theory: classical, pre-quantum, quantum, perturbative quantum

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Integration theory

Contents

Idea

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).

Definition

For graded-commutative algebras

Definition

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.

Definition

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).

Remark

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)

and

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

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.