Uniqueness of product and coproduct decompositions in rational homotopy theory

Abstract:

Let $X$ be a nilpotent rational homotopy type such that (1) $S(X)$, the image of the Hurewicz map has finite total rank, and (2) the basepoint map of $M$, a minimal algebra for $X$, is an element of the Zariski closure of ${\text {Aut}}(M)$ in ${\text {End}}(M)$ (i.e. $X$ has "positive weight"). Then (A) any retract of $X$ satisfies the two properties above, (B) any two irreducible product decompositions of $X$ are equivalent, and (C) any two irreducible coproduct decompositions of $X$ are equivalent.