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arXiv · 1805.02001

Isomorphism problem and homological properties of DG free algebras

X. -F. Mao·J. -F. Xie·Y. -N. Yang·Almire. Abla
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Abstract

A differential graded (DG for short) free algebra $\mathcal{A}$ is a connected 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 differential structures on DG free algebras are in one to one correspondence with the set of crisscross ordered $n$-tuples of $n\times n$ matrixes. We also give a criterion to judge whether two DG free algebras are isomorphic. As an application, we consider the case of $n=2$. Based on the isomorphism classification, we compute the cohomology graded algebras of non-trivial DG free algebras with $2$ generators, and show that all those non-trivial DG free algebras are Koszul and Calabi-Yau.

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X. -F. Mao, J. -F. Xie, Y. -N. Yang, Almire. Abla. 2018-05-05. Isomorphism problem and homological properties of DG free algebras. https://arxiv.org/abs/1805.02001

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