Exact Hochschild extensions and deformed Calabi-Yau completions

TL;DR

This paper explores Hochschild extensions of dg algebras, establishing their symmetry properties and linking their Koszul duals to Calabi-Yau and deformed Calabi-Yau completions, advancing understanding of algebraic dualities.

Contribution

It introduces Hochschild extensions as $A_ abla$-algebras, proves their symmetry in the finite-dimensional case, and connects their Koszul duals to Calabi-Yau completions.

Findings

All exact Hochschild extensions are symmetric.

Koszul dual of trivial extension is Calabi-Yau completion.

Koszul dual of exact Hochschild extension is deformed Calabi-Yau completion.

Abstract

We introduce the Hochschild extensions of dg algebras, which are $A_\infty$-algebras. We show that all exact Hochschild extensions are symmetric Hochschild extensions, more precisely, every exact Hochschild extension of a finite dimensional complete typical dg algebra is a symmetric $A_\infty$-algebra. Moreover, we prove that the Koszul dual of trivial extension is Calabi-Yau completion and the Koszul dual of exact Hochschild extension is deformed Calabi-Yau completion, more precisely, the Koszul dual of the trivial extension of a finite dimensional complete dg algebra is the Calabi-Yau completion of its Koszul dual, and the Koszul dual of an exact Hochschild extension of a finite dimensional complete typical dg algebra is the deformed Calabi-Yau completion of its Koszul dual.