Homotopy Dg Algebras Induce Homotopy Bv Algebras

Let TA denote the space underlying the tensor algebra of a vector space A. In this short note, we show that if A is a differential graded algebra, then TA is a differential BatalinVilkovisky algebra. Moreover, if A is an A∞ algebra, then TA is a commutative BV∞ algebra. 1. Main Statement Let (A, dA) be a complex over a commutative ring R. Our convention is that dA is of degree +1. The space TA = ⊕ n>0 A ⊗n is graded by declaring monomials of homogeneous elements a1 ⊗ · · · ⊗ an ∈ A⊗n to be of degree |a1|+ · · ·+ |an|+ n. There is a shuffle product • : TA⊗ TA → TA generated by (a1 ⊗ · · · ⊗ an) • (an+1 ⊗ · · · ⊗ an+m) := ∑ σ∈S(n,m) (−1)κ · aσ−1(1) ⊗ · · · ⊗ aσ−1(n+m), where S(n,m) is the set of all (n,m)-shuffles, i.e. S(n,m) is the set of all permutations σ ∈ Σn+m with σ(1) < · · · < σ(n) and σ(n + 1) < · · · < σ(n + m), (cf. [6]). Here (−1)κ is the Koszul sign, which introduces a factor of (|ai|+1)(|aj |+1) whenever the elements ai and aj move past one another in a shuffle. Note that for degree zero elements of A, this Koszul sign is just sgn(σ), the sign of the permutation σ. The shuffle product makes TA into a graded commutative associative algebra. Recall that TA is also a coalgebra under the usual tensor coproduct. There is a differential d : TA → TA (of degree +1) given by extending the differential dA : A → A as a coderivation of the tensor coproduct, see e.g. [7]: d(a1 ⊗ · · · ⊗ an) = n ∑ i=0 (−1)|a1|+···+|ai−1|+i−1a1 ⊗ · · · ⊗ dA(ai)⊗ · · · ⊗ an The second author was partially supported by the Max-Planck Institute in Bonn, Germany. The third author was supported in part by a grant from The City University of New York PSC-CUNY Research Award Program. We would like to thank Gabriel Drummond-Cole and Bruno Vallette for useful discussions about BV∞ algebras. Received December 21, 2010, revised March 02, 2011; published on July 12, 2011. 2000 Mathematics Subject Classification: 16E45, 17B60, 18G55.