On the Indecomposable Elements of the Bar Construction

An explicit formula for a canonical splitting <!-- MATH $s:Q\mathcal{B}({\mathcal{E}^ \cdot }) \to \mathcal{B}({\mathcal{E}^ \cdot })$ --> of the projection <!-- MATH $\mathcal{B}({\mathcal{E}^ \cdot }) \to Q\mathcal{B}({\mathcal{E}^ \cdot })$ --> of the bar construction on a commutative d.g. algebra <!-- MATH ${\mathcal{E}^\cdot}$ --> onto its indecomposables is given. We prove that induces a d.g. algebra isomorphism <!-- MATH $\Lambda (Q\mathcal{B}({\mathcal{E}^\cdot})) \to \mathcal{B}({\mathcal{E}^\cdot})$ --> and that <!-- MATH $H(Q\mathcal{B}({\mathcal{E}^\cdot}))$ --> is isomorphic with <!-- MATH $QH(\mathcal{B}({\mathcal{E}^\cdot}))$ --> .