Isomorphism problem and homological properties of DG free algebras

TL;DR

This paper classifies differential graded free algebras by their differential structures, provides criteria for isomorphism, and analyzes their cohomology, demonstrating that certain classes are Koszul and Calabi-Yau.

Contribution

It establishes a one-to-one correspondence between differential structures and matrix tuples, and classifies isomorphisms of DG free algebras with two generators.

Findings

Differential structures correspond to crisscross ordered matrix tuples.

Criteria for isomorphism of DG free algebras are provided.

All non-trivial DG free algebras with two generators are Koszul and Calabi-Yau.

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.