Higher-Dimensional Algebra VI: Lie 2-Algebras

Higher-Dimensional Algebra VI: Lie 2-Algebras

[John Baez]

Here's a new paper that studies categorified Lie algebras:

Higher-Dimensional Algebra VI: Lie 2-algebras
John C. Baez and Alissa S. Crans

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 Vect 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 linear 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 paper will soon appear on the mathematics arXiv, but
the PDF version on my website looks a tiny bit better:

http://math.ucr.edu/home/baez/hda6.pdf