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A differential graded (DG for short) free algebra $\mathcal{A}$ is aconnected cochain DG algebra such that its underlying graded algebra is$$\mathcal{A}^{\#}=\k\langle x_1,x_2,\cdots, x_n\rangle,\,\, \text{with}\,\,|x_i|=1,\,\, \forall i\in \{1,2,\cdots, n\}.$$ We prove that the differentialstructures on DG free algebras are in one to one correspondence with the set ofcrisscross ordered $n$-tuples of $n\times n$ matrixes. We also give a criterionto judge whether two DG free algebras are isomorphic.  As an application, we consider the case of $n=2$. Based on the isomorphismclassification, we compute the cohomology graded algebras of non-trivial DGfree algebras with $2$ generators, and show that all those non-trivial DG freealgebras are Koszul and Calabi-Yau.

Isomorphism problem and homological properties of DG free algebras