Gerstenhaber Bracket on Double Hochschild Complex and Deformation Theory

We construct a differential and a Lie bracket on the space {Hom(A, A)},k,l≥0 for any associative algebra A. The restriction of this bracket to the space {Hom(A, A)},k≥0 is exactly the Gerstenhaber bracket. We discuss some formality conjecture related with this construction. We also discuss some applications to deformation theory. 0. There exists a well-known way (due to Jim Stasheff) to define the Hochschild differential and the Gerstenhaber bracket via coderivations on the cofree coalgebra cogenerated by the vector space A[1] (A is an associative algebra). Analogously, one can define a dual differential and bracket on the graded space {Hom(A,A)},k≥0 using derivations of tensor algebra, generated by the space A[1] for any coalgebra A. In both cases the bracket does not depend on the (co)algebra structure, and only the differential does. In these notes we generalize this construction involving all “differential operators” on the tensor algebra T (A[1]), not only of the first order. In such a way, we define a bidifferential and a bracket on the bigraded space {Hom(A, A)},k,l≥0. It turns out that for any associative algebra A with unit the total cohomology of the bicomplex are equal examples to zero. So, the interesting examples appear for the algebras without unit, for example, for the algebras of polynomials without unit. 1. Let V be a vector space, and let Ψ1 : V ⊗k1 → V ⊗l1 , Ψ2 : V ⊗k2 → V ⊗l2 be any two maps. We are going to define the bracket [Ψ1,Ψ2]. Let Ψ: V ⊗k → V ⊗l be any map. It defines a map i(Ψ): V ⊗N → V ⊗(N−k+l), N ≫ 0, as follows: (1) i(Ψ)(v1 ⊗ v2 ⊗ . . .⊗ vn) = Ψ(v1 ⊗ . . .⊗ vk)⊗ vk+1 ⊗ . . .⊗ vN + + v1 ⊗Ψ(v2 ⊗ . . .⊗ vk+1)⊗ . . . ⊗ vN + . . .+ + v1 ⊗ v2 ⊗ . . . ⊗Ψ(vN−k+1 ⊗ . . .⊗ vN ). It is clear that the composition i(Ψ1) ◦ i(Ψ2) has not the form i(Ψ3) for some Ψ3. Nevertheless, one has the following statement. Lemma. The bracket i(Ψ1) ◦ i(Ψ2)− i(Ψ2) ◦ i(Ψ1) has a form i (