Deformation theory with homotopy algebra structures on tensor products

TL;DR

This paper develops new homotopy algebra structures on tensor products and mapping spaces, extending classical algebraic structures in deformation theory and enabling advanced analysis of algebra morphisms.

Contribution

It introduces natural homotopy Lie and associative algebra structures on tensor products and mapping spaces, compatible with homotopy theory concepts.

Findings

Constructed deformation complexes for algebra morphisms

Represented deformation ∞-groupoids for dg Lie algebras

Extended algebraic structures via Manin products of operads

Abstract

In order to solve two problems in deformation theory, we establish natural structures of homotopy Lie algebras and of homotopy associative algebras on tensor products of algebras of different types and on mapping spaces between coalgebras and algebras. When considering tensor products, such algebraic structures extend the Lie algebra or associative algebra structures that can be obtained by means of the Manin products of operads. These new homotopy algebra structures are proven by to compatible with the concepts of homotopy theory: $\infty$-morphisms and the Homotopy Transfer Theorem. We give a conceptual interpretation of their Maurer-Cartan elements. In the end, this allows us to construct the deformation complex for morphisms of algebras over an operad and to represent the deformation $\infty$-groupoid for differential graded Lie algebras.