Misc,
Lie 2-Algebras
(2004)cite arxiv:math/0409602 Comment: Ph.D. Thesis; 119 pages LaTeX with diagram.sty and xypic diagrams.
Abstract
We categorify the theory of Lie algebras beginning with a new notion of
categorified vector space, or `2-vector space', which we define as an internal
category in Vect, the category of vector spaces. We then define a `semistrict
Lie 2-algebra' to be a 2-vector space 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. Much of the content of this first chapter has
already appeared in a separate paper coauthored with John Baez, Higher
Dimensional Algebra VI: Lie 2-algebras.
We then explore the relationship between Lie algebras and algebraic
structures called `quandles'. A quandle is a set equipped with two binary
operations satisfying axioms that capture the essential properties of the
operations of conjugation in a group and algebraically encode the three
Reidemeister moves. Indeed, we describe the relation to groups and show that
quandles give invariants of braids. We further show that both Lie algebras and
quandles give solutions of the Yang--Baxter equation, and explain how
conjugation plays a prominent role in the both the theories of Lie algebras and
quandles. Inspired by these commonalities, we provide a novel, conceptual
passage from Lie groups to Lie algebras using the language of quandles.
Moreover, we propose relationships between higher Lie theory and
higher-dimensional braid theory. We conclude with evidence of this connection
by proving that any semistrict Lie 2-algebra gives a solution of the
Zamolodchikov tetrahedron equation, which is the higher-dimensional analog of
the Yang--Baxter equation.
