Strongly homotopy Lie algebras

TL;DR

This paper explores the structure and properties of strongly homotopy Lie algebras, extending classical Lie algebra concepts to the homotopy setting and examining their universal enveloping algebras and monoidal structures.

Contribution

It demonstrates the homotopy analog of classical Lie-associative algebra relations and investigates the universal enveloping algebra functor within this framework.

Findings

Established the monoidal structure of homotopy associative algebras.

Showed the universal enveloping algebra as a unital coassociative cocommutative coalgebra.

Analyzed the relation between homotopy modules and weak homotopy maps.

Abstract

The present paper can be thought of as a continuation of the paper "Introduction to sh Lie algebras for physicists" by T. Lada and J. Stasheff (International Journal of Theoretical Physics Vol. 32, No. 7 (1993), 1087--1103, appeared also as preprint hep-th/9209099) which provided an exposition of the basic ingredients of the theory of strongly homotopy Lie algebras sufficient for the underpinnings of the physically relevant examples. We demonstrate the `strong homotopy' analog of the usual relation between Lie and associative algebras and investigate the universal enveloping algebra functor emerging as the left adjoint of the symmetrization functor. We show that the category of homotopy associative algebras carries a natural monoidal structure such that the universal enveloping algebra is a unital coassociative cocommutative coalgebra with respect to this monoidal structure. The last section is concerned with the relation between homotopy modules and weak homotopy maps. The present paper is complementary to what currently exists in the literature, both physical and mathematical.