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Algebraic string operations. (English) Zbl 1144.55012
It is well-known that the Hochschild complex $$C^*(A,A)$$ of an associative algebra $$A$$ possesses a rich algebraic structure. Deligne’s conjecture (proved by several people) states that the operad of chains on the little disc operad acts on $$C^*(A,A)$$. The discovery of string topology by M. Chas and D. Sullivan suggests that if an $$A_\infty$$-algebra possesses a kind of Poincaré duality, richer algebraic operations should act on $$C^*(A,A)$$. The present authors made a significant progress in this direction.
Namely, they offer a definition of a $$V_k$$-algebra for all $$1\leq k\leq \infty$$. This concept is homotopy invariant. If $$k=1$$ then $$V_1$$-algebras are precisely $$A_{\infty}$$-algebras. For $$k=2$$ $$V_2$$-algebras are $$A_\infty$$-algebras with homotopy co-inner product. For $$k\geq 3$$ the concept is new and quite interesting and is related to open strings. Moreover using directed graphs they construct a sequence of inclusions of PROPs $$\text{DG}_1\subset \text{DG}_2 \subset \cdots \subset \text{DG}_\infty$$ in such a way that if $$A$$ is a $$V_k$$-algebra then then $$C^*(A,A)$$ is an algebra over $$\text{DG}_k$$, $$1\leq k\leq \infty$$.

##### MSC:
 55P48 Loop space machines and operads in algebraic topology 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) 18D50 Operads (MSC2010)
##### Keywords:
string topology; Hochschild cohomology
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##### References:
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