A reduction of the string bracket to the loop product

The negative cyclic homology for a differential graded algebra over the rational field has a quotient of the Hochschild homology as a direct summand if the S-action is trivial. With this fact, we show that the string bracket in the sense of Chas and Sullivan is reduced to the loop product followed by the BV operator on the loop homology provided the given manifold is BV-exact. . The reduction is indeed derived from the equivalence between the BV-exactness and the triviality of the S-action. Moreover, it is proved that a Lie bracket on the loop cohomology of the classifying space of a connected compact Lie group possesses the same reduction. By using these results, we consider the nontriviality of string brackets. We also show that a simply connected space with positive weights is BV-exact. Furthermore, the higher BV-exactness is discussed featuring the cobar-type Eilenberg-Moore spectral sequence.