Strongly homotopy Lie algebras
Tom Lada, Martin Markl
TL;DR
This work extends the theory of strongly homotopy Lie algebras by building a parallel SH associative framework (A(m)) and establishing a robust Lie–associative correspondence in characteristic zero. It introduces the L(m) and A(m) formalisms, characterizes L(m) via degree $-1$ coderivations on cofree cocommutative coalgebras, and proves a skew-symmetrization relation that yields SH Lie algebras from SH associative algebras. The universal enveloping construction ${\cal U}_m$ is developed as a left adjoint to the symmetrization functor, with ${\cal U}_m(L)$ carrying a natural unital cocommutative coalgebra structure in a strict monoidal framework on ${\bf A}(m)$. The paper also develops a theory of L(m)-modules and weak L(m)-maps, connecting module structures to differential graded Lie algebra morphisms and expanding the landscape of higher homotopy algebra applicable to deformation theory and mathematical physics.
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.