nLab
Deligne conjecture

Context

Higher algebra

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

Statement

The original Deligne conjecture on the structure of Hochschild cohomology of an associative algebra and more generally an A-∞ algebra states that Hochschild cohomology naturally has the structure of a BV-algebra and that moreover the full Hochschild cohomology complex has the structure of an algebra over an operad over the (framed) little disk operad. (The second statement implies the first since (BV-)Gerstenhaber algebras are algebras over an operad over the (framed) little disk operad.)

Proofs of the conjecture have been given first by McLure-Smith and others, then also by Kontsevich-Soibelman, and Tamarkin.

A proof for general Ek-algebras in symmetric monoidal (∞,1)-categories is in (Lurie, section 2.5).

Geometric interpretation

A general geometric (higher geometric) interpretation has been indicated in Ben-ZviFrancisNadler. They observed (see Hochschild cohomology for details) that the Hochschild homology of 𝒪(X)\mathcal{O}(X) is naturally interpreted as the algebra of functions on the derived loop space X\mathcal{L}X. In the presence of good geometric ∞-function theory this naturally induces an action of the little disk operad on 𝒪(X)\mathcal{O}(\mathcal{L}X). Since a Gerstenhaber algebra is an algebra over the homology of the little disk operad, this immediately explains the existence of this structure.

For higher algebras and higher monoidal categories

More generally, analogs of the statement of the Deligne conjecture exist and work for (,n)(\infty,n)-algebras: k-tuply monoidal n-categories. This is closely related to the statement (and proofs) of the delooping hypothesis.

This case is discussed in (Francis) and (Lurie).

In (KockToen) it is shown that in a monoidal model category that is also a simplicial model category the derived hom-space Hom(I,I)\mathbb{R} Hom(I,I) from the tensor unit to itself is a E-2 algebra.

History

Historically it was first found that there is the structure of a Gerstenhaber algebra on HH (A,A)HH^\bullet(A,A). By (Cohen) it was known that Gerstenhaber algebras arise as the homology of E2-algebras in chain complexs. In a letter in 1993 Deligne wondered whether the Gerstenhaber algebra structure on the Hochschild cohomology HH (A,A)HH^\bullet(A,A) lifts to an E2-algebra-structure on the cochain complex C (A,A)C^\bullet(A,A).

In GerstenhaberVoronov (1994) a resolution of the Gerstenhaber algebra structure was given, but the relationship to E 2E_2-algebras remaind unclear.

In (Tamarkin (1998)) a genuine resolution in the model structure on operads of the Gerstenhaber operad was given and shown to act via the Gerstenhaber-Voronov construction on C (A,A)C^\bullet(A,A). This proved Deligne’s conjecture.

Various authors later further refined this result. A summary of this history can be found in (Hess).

In Hu-Kriz-Voronov (2003) it was further shown that for AA an En-algebra, C (A,A)C^\bullet(A,A) is an E n+1E_{n+1}-algebra.

References

Direct proofs of the Deligne conjecture have been given in.

  • McLure-Smith,
  • Tamarkin,

A review is in

  • Cohen
  • Gertstenhaber-Voronov
  • Hu, Kriz, Voronov

A transparent higher geometric interpretation in a suitably dualizable context is indicated in

Section 2.5 of

A construction in monoidal model categories is in

  • Joachim Kock, Bertrand Toen, Simplicial localization of monoidal structures, and a non-linear version of Deligne’s conjecture Compositio Math. 141 (2005), 253-261 (arXiv)

Revised on March 11, 2011 21:44:12 by David Roberts (202.124.89.49)