The de Rham theorem for the noncommutative complex of Cenkl and Porter

Abstract

We use noncommutative differential forms (which were first introduced by Connes) to construct a noncommutative version of the complex of Cenkl and PorterΩ∗,∗(X)for a simplicial setX. The algebraΩ∗,∗(X)is a differential graded algebra with a filtrationΩ∗,q(X)⊂Ω∗,q+1(X), such thatΩ∗,q(X)is aℚq-module, whereℚ0=ℚ1=ℤandℚq=ℤ[1/2,…,1/q]forq>1. Then we use noncommutative versions of the Poincaré lemma and Stokes' theorem to prove the noncommutative tame de Rham theorem: ifXis a simplicial set of finite type, then for eachq≥1and anyℚq-moduleM, integration of forms induces a natural isomorphism ofℚq-modulesI:Hi(Ω∗,q(X),M)→Hi(X;M)for alli≥0. Next, we introduce a complex of noncommutative tame de Rham currentsΩ∗,∗(X)and we prove the noncommutative tame de Rham theorem for homology: ifXis a simplicial set of finite type, then for eachq≥1and anyℚq-moduleM, there is a natural isomorphism ofℚq-modulesI:Hi(X;M)→Hi(Ω∗,q(X),M)for alli≥0.

The Nakayama Automorphism of the Almost Calabi-Yau Algebras Associated to SU(3) Modular Invariants

“…Since A is an augmented S-algebra, i.e. A 0 = S and there is an augmentation ϕ : A → S such that ϕ(1) = 1, by the non-commutative Poincaré lemma [27] (see also [30,Lemma 4.5]), HDR n (A) = HDR n (S) = 0 for all n. Thus, from [16, Lemma 3.6.1], there is an exact sequence…”