| Higher-dimensional algebra VI: Lie 2-algebras. Theory and Applications of Categories (2004) | |||||||||||||||
Abstract | |||||||||||||||
| , *] : L*L! L satisfying the Jacobi identity up to a completely antisymmetric trilinearnatural 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, andindeed 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 L1-algebras in the sense of Stasheff. We also study strict and skeletal Lie2-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. Thisclassification allows us to construct for any finite-dimensional Lie algebra g a canonical1-parameter family of Lie 2-algebras g ~ which reduces to g at ~ = 0. These are closelyrelated to the 2-groups G ~ constructed in a companion paper. 1. Introduction One of the goals of higher-dimensional algebra is to `categorify ' mathematical concepts, replacing equational laws by isomorphisms satisfying new coherence laws of their own. By iterating this process, we hope to find n-categorical and eventually!-categorical generalizations of as many mathematical concepts as possible, and use these to strengthen--and often simplify--the connections between different parts of mathematics. The previous paper of this series, HDA5 [6], categorified the concept of Lie group and began to explore the resulting theory of `Lie 2-groups'. Here we do the same for the concept of Lie algebra, obtaining a theory of `Lie 2-algebras'. | |||||||||||||||
Publication details | |||||||||||||||
| |||||||||||||||