Algebraic string operations
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 \(\text{DG}_k\), \(1\leq k\leq \infty\).
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