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HigherDimensional Algebra VI: Lie 2Algebras  John C. Baez
; Alissa S. Crans
;  Date: 
19 Jul 2003  Subject:  Quantum Algebra; Category Theory  math.QA math.CT  Abstract:  The theory of Lie algebras can be categorified starting from a new notion of "2vector space", which we define as an internal category in Vect. There is a 2category 2Vect having these 2vector spaces as objects, "linear functors" as morphisms and "linear natural transformations" as 2morphisms. We define a "semistrict Lie 2algebra" to be a 2vector space L 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. This law is closely related to the Zamolodchikov tetrahedron equation, and indeed we prove that any semistrict Lie 2algebra gives a solution of this equation, just as any Lie algebra gives a solution of the YangBaxter equation. We construct a 2category of semistrict Lie 2algebras and prove that it is 2equivalent to the 2category of 2term Linfinity algebras in the sense of Stasheff. We also study strict and skeletal Lie 2algebras, obtaining the former from strict Lie 2groups and using the latter to classify Lie 2algebras in terms of 3rd cohomology classes in Lie algebra cohomology. This classification allows us to construct for any finitedimensional Lie algebra g a canonical 1parameter family of Lie 2algebras g_hbar which reduces to g at hbar = 0. These are closely related to the 2groups G_hbar constructed in a companion paper.  Source:  arXiv, math.QA/0307263  Services:  Forum  Review  PDF  Favorites 


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