The BV Algebra on Hochschild Cohomology Induced by Infinity Inner Products

We define a BV-structure on the Hochschild-cohomology of a unital, associative algebra A with a symmetric, invariant and non-degenerate inner product. The induced Gerstenhaber algebra is the one described in Gerstenhabers original paper on Hochschild-cohomology. We also prove the corresponding theorem in the homotopy case, namely we define the BV-structure on the Hochschild-cohomology of a unital A-infinity-algebra with a symmetric and non-degenerate infinity-inner product.