El Infinity

On Weak Lie 3-Algebras

Strongly homotopy Lie algebras or $\mathrm{L}_\infty$-algebras are a generalization of (dg) Lie algebras with better homotopy properties. In particular, a version of the homotopy transfer theorem holds. This is, however, only true when working over a field of characteristic $0.$ When working over an arbitrary unital commutative ring, one must generalize the definition even further, relaxing the skew-symmetry of the brackets up to homotopy in addition to the Jacobi identity.

I just finished my article “On weak Lie $3$-algebras,” in which I consider precisely this problem. I describe a general step by step approach and give explicit definitions for the case of underlying $3$-term complexes. In this post, I’ll try to give a rough idea of the approach used and a quick overview of the main results.