Versal deformation theory of algebras over a quadratic operad

TL;DR

This paper develops a deformation theory framework for algebras over quadratic operads, introducing a versal deformation that encapsulates all possible deformations over complete local algebras.

Contribution

It introduces a comprehensive deformation theory for quadratic operad algebras and constructs a versal deformation capturing all deformations.

Findings

Established a deformation theory for quadratic operad algebras.

Constructed a versal deformation inducing all other deformations.

Provided a method to analyze algebra deformations over local algebras.

Abstract

We develop deformation theory of algebras over quadratic operads where the parameter space is a commutative local algebra. We also give a construction of a distinguised deformation of an algebra over a quadratic operad with a complete local algebra as its base--the so called `versal deformation'--which induces all other deformations of the given algebra.