Higher-Dimensional Algebra VI: Lie 2-Algebras
John C. Baez, Alissa S. Crans
TL;DR
The paper develops a categorified analogue of Lie algebras by introducing semistrict Lie 2-algebras realized as 2-vector spaces with a bilinear skew bracket and a Jacobiator whose coherence is governed by a Jacobiator identity. It establishes a 2-category of these structures and proves a 2-equivalence with the 2-category of 2-term $L_\infty$-algebras, linking categorified Lie theory with homological algebra and topology via the Zamolodchikov tetrahedron equation. It then analyzes strict and skeletal variants, showing skeletal Lie 2-algebras are classified by data $(\mathfrak{g},V,\rho,[l_3])$ with $[l_3]\in H^3(\mathfrak{g},V)$ and constructing canonical deformations $\mathfrak{g}_\hbar$; these results illuminate deep connections between categorified Lie theory, representation theory, and cohomology. The work advances a coherent program to integrate Lie 2-algebras into a broader theory of $n$-groups and higher algebra with potential ties to quantum groups and higher-dimensional topology.
Abstract
The theory of Lie algebras can be categorified starting from a new notion of "2-vector space", which we define as an internal category in Vect. There is a 2-category 2Vect having these 2-vector spaces as objects, "linear functors" as morphisms and "linear natural transformations" as 2-morphisms. We define a "semistrict Lie 2-algebra" to be a 2-vector space L equipped with a skew-symmetric bilinear functor satisfying the Jacobi identity up to a completely antisymmetric trilinear natural transformation called the "Jacobiator", which in turn must satisfy a certain law of its own. This law is closely related to the Zamolodchikov tetrahedron equation, and indeed we prove that any semistrict Lie 2-algebra gives a solution of this equation, just as any Lie algebra gives a solution of the Yang-Baxter equation. We construct a 2-category of semistrict Lie 2-algebras and prove that it is 2-equivalent to the 2-category of 2-term L-infinity algebras in the sense of Stasheff. We also study strict and skeletal Lie 2-algebras, obtaining the former from strict Lie 2-groups and using the latter to classify Lie 2-algebras in terms of 3rd cohomology classes in Lie algebra cohomology. This classification allows us to construct for any finite-dimensional Lie algebra g a canonical 1-parameter family of Lie 2-algebras g_hbar which reduces to g at hbar = 0. These are closely related to the 2-groups G_hbar constructed in a companion paper.