Lie 2Algebras
Abstract
We categorify the theory of Lie algebras beginning with a new notion of categorified vector space, or `2vector space', which we define as an internal category in Vect, the category of vector spaces. We then define a `semistrict Lie 2algebra' to be a 2vector space equipped with a skewsymmetric 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 2algebras. 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 YangBaxter 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 higherdimensional braid theory. We conclude with evidence of this connection by proving that any semistrict Lie 2algebra gives a solution of the Zamolodchikov tetrahedron equation, which is the higherdimensional analog of the YangBaxter equation.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 September 2004
 arXiv:
 arXiv:math/0409602
 Bibcode:
 2004math......9602C
 Keywords:

 Quantum Algebra;
 Category Theory;
 Geometric Topology
 EPrint:
 Ph.D. Thesis