The complex of formal operations on the Hochschild chains of commutative algebras

Abstract

Abstract. We compute the homology of the complex of formal operations on the Hochschild complex of differential graded commutative algebras as defined by Wahl and prove that these can be built as infinite sums of operations obtained from Loday's shuffle operations, Connes' boundary operator and the shuffle product.

Hochschild homology of structured algebras

“…Doing this we recover the same formuli as in the local-to-global approach. Our construction can be seen as a formalization and extension of the method of [11] which considered the case of A ∞ -Frobenius algebras.Our main result, which we will explain now in more details, gave rise to new computations, including a complete description of the operations on the Hochschild complex of commutative algebras [35], a description of a large complex of operations on the Hochschild complex of commutative Frobenius algebras [34] and a description of the universal operations given any type of algebra [61].An A ∞ -algebra can be described as an enriched symmetric monoidal functor from a certain dg-category A ∞ to Ch, the dg-category of chain complexes over Z. The category A ∞ is what is called a dg-prop, a symmetric monoidal dg-category with objects the natural numbers.…”

“…Our main result, which we will explain now in more details, gave rise to new computations, including a complete description of the operations on the Hochschild complex of commutative algebras [35], a description of a large complex of operations on the Hochschild complex of commutative Frobenius algebras [34] and a description of the universal operations given any type of algebra [61].…”

“…(The prop of open-closed Sullivan diagrams defined in [30] has its closed part isomorphic to the Sullivan diagrams considered here, see Remark 2.15.) Further applications of our methods in the case of commutative and commutative Frobenius algebras where obtained by Klamt in [35,34]. Other interesting examples of families of algebras to consider would be algebras over Kaufmann's prop of open Sullivan diagrams [30] (see Section 6.7), Hopf algebras, Poisson algebras and E n -algebras, to name a few.…”

Universal operations in Hochschild homology

“…We use the same spectral sequence to identify the homology of Nat E ([ n 1 m 1 ], [ n 2 m 2 ]) in three cases detailed below: the case of unital A ∞ -algebras, the case of Frobenius algebras, and the case of open field theories. Klamt computed in addition the case of commutative algebras in [17], and, to a large extend, the case of commutative Frobenius algebras in [19]. To further exemplify our approach, we show in Proposition 2.14 how the cap product in Hochschild homology can be seen as part of the chain complex of formal operations Nat E ([ 1 0 ], [ 1 0 ]) for E = End(A) the endomorphism prop of the algebra A considered.…”

Natural operations on the Hochschild complex of commutative Frobenius algebras via the complex of looped diagrams

“…On the other hand, every commutative Frobenius dg-algebra is of course a differential graded commutative algebra. In [Kla13] we give a description of the homology of all formal operations for differential graded commutative algebras in terms of Loday's shuffle operations (defined in [Lod89]) and the Connes' boundary operator. Well-known operations which are covered in this complex are Loday's λ-operations and the shuffle product C * (A, A) ⊗ C * (A, A) → C * (A, A).…”