The odd pairing of cyclic cohomology with invertibles #
Let A be a unital complex algebra and M a complex vector space. For an invertible u of A
and a cyclic M-valued cocycle φ of odd degree n = 2m + 1, Connes' odd pairing is
⟨[u], [φ]⟩ = (1 + i)⁻¹ 2⁻ⁿ Γ(n/2 + 1)⁻¹ φ(u⁻¹ - 1, u - 1, …, u⁻¹ - 1, u - 1),
with n + 1 alternating arguments. This file shows that the value depends only on the cyclic
cohomology class of φ, and constructs the resulting linear map HC²ᵐ⁺¹(A; M) → M. Composed with
matrix extension HC²ᵐ⁺¹(A; M) → HC²ᵐ⁺¹(Matrix ι ι A; M), it pairs invertible matrices with cyclic
cohomology through φ#Tr. Its homotopy invariance and multiplicativity, the relations through
which it descends to algebraic K₁, are proved in
TauCeti.RingTheory.CyclicCohomology.OddPairing.Invariance.
The pairing descends to cohomology because the entries x = u⁻¹ - 1 and y = u - 1 are
quasi-inverse to each other in both orders: y + x + xy = 0 = x + y + yx. More generally, if
cyclically adjacent entries of a tuple (x₀, …, xₙ₊₁) are quasi-inverse, then every Hochschild
face xᵢxᵢ₊₁ = -(xᵢ + xᵢ₊₁) splits into the two tuples deleting xᵢ₊₁ or xᵢ. For a cyclic
cochain ψ the alternating sum of these terms telescopes, so (bψ)(x₀, …, xₙ₊₁) = 0
(TauCeti.hochschildCoboundary_apply_eq_zero_of_add_add_mul_eq_zero).
Main definitions #
TauCeti.cyclicCohomology.oddPairing: the pairingHC²ᵐ⁺¹(A; M) → Mwith an invertible.
Main results #
TauCeti.cyclicCohomology.oddPairing_mk: the pairing evaluated on the class of a cocycle.TauCeti.cyclicCohomology.oddPairing_matrixExtension_mk: Connes' formula⟨[u], [φ]⟩ = (1 + i)⁻¹ 2⁻ⁿ Γ(n/2 + 1)⁻¹ (φ#Tr)(u⁻¹ - 1, u - 1, …, u⁻¹ - 1, u - 1)for an invertible matrixu.TauCeti.cyclicCohomology.oddPairing_matrixExtension_of_val_eq_fromBlocks: the pairing is additive on direct sums,⟨[u ⊕ v], [φ]⟩ = ⟨[u], [φ]⟩ + ⟨[v], [φ]⟩.TauCeti.cyclicCohomology.oddPairing_matrixExtension_of_val_eq_fromBlocks_one: the pairing is unchanged by stabilizationu ↦ u ⊕ 1.Equiv.oddPairing_reindexGL: the pairing is unchanged by relabelling the matrix indices.TauCeti.cyclicCohomology.oddPairing_map: naturality of the pairing for algebra homomorphisms.
References #
- A. Connes, Noncommutative Geometry, Chapter III, §3.
- J.-L. Loday, Cyclic Homology, §8.3.
A cyclic coboundary of odd degree 2m + 1 vanishes on the tuple
(u⁻¹ - 1, u - 1, …, u⁻¹ - 1, u - 1) of an invertible u.
Connes' odd pairing ⟨[u], [φ]⟩ = (1 + i)⁻¹ 2⁻ⁿ Γ(n/2 + 1)⁻¹ φ(u⁻¹ - 1, u - 1, …, u - 1)
of an invertible u with cyclic cohomology classes of odd degree n = 2m + 1.
Instances For
The odd pairing of an invertible u with the class of a cyclic cocycle φ of degree
n = 2m + 1 is (1 + i)⁻¹ 2⁻ⁿ Γ(n/2 + 1)⁻¹ φ(u⁻¹ - 1, u - 1, …, u⁻¹ - 1, u - 1).
The odd pairing of the unit vanishes.
Naturality of the odd pairing: the image of an invertible u under an algebra
homomorphism f : A →ₐ[ℂ] B pairs with a class of B as u pairs with the pullback of the class
along f.
Connes' odd pairing of an invertible matrix u with cyclic cohomology: on the class of a
cyclic cocycle φ of degree n = 2m + 1, it is
(1 + i)⁻¹ 2⁻ⁿ Γ(n/2 + 1)⁻¹ (φ#Tr)(u⁻¹ - 1, u - 1, …, u⁻¹ - 1, u - 1).
Additivity of the odd pairing on direct sums: the odd pairing of an invertible matrix w
which is block diagonal with diagonal blocks u and v is the sum of the odd pairings of u
and v.
Stabilization of the odd pairing: enlarging an invertible matrix u to the block diagonal
matrix u ⊕ 1 does not change its odd pairing.
Relabelling invariance of the odd pairing: reindexing the rows and columns of an invertible matrix along an equivalence of index types does not change its odd pairing.