B ∞-Algebra Structure in Homology of a Homotopy Gerstenhaber Algebra

Abstract

The minimality theorem states, in particular, that on cohomology H(A) of a dg algebra there exists sequence of operations mi : H(A)⊗i → H(A), i = 2, 3, . . . , which form a minimal A ∞ -algebra (H(A), {m i }). This structure defines on the bar construction BH(A) a correct differential dm so that the bar constructions (BH(A), d m ) and BA have isomorphic homology modules. It is known that if A is equipped additionally with a structure of homotopy Gerstenhaber algebra, then on BA there is a multiplication which turns it into a dg bialgebra. In this paper, we construct algebraic operations Ep,q : H(A) ⊗p ⊗H(A) ⊗q → H(A), p, q = 0, 1, 2, . . ., which turn (H(A), {m i }, {E p,q }) into a B ∞ -algebra. These operations determine on BH(A) correct multiplication, so that (BH(A), d m ) and BA have isomorphic homology algebras.