Higher-Dimensional Algebra VI: Lie 2-Algebras

By John C. Baez, Alissa S. Crans. Theory and Applications of Categories, 2003

436 citations recorded by the source corpus.

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 2Vect 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 (·, · ): L×L → L 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 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∞-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 classification allows us to construct for any finite-dimensional Lie algebra g a canonical 1-parameter family of Lie 2-algebras g which reduces to g at � = 0. These are closely related to the 2-groups G constructed in a companion paper.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fdba60bdc761c98659e8ed5ba08458ecb5224827

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Citations in the corpus

Cohomology of Categorical Self-Distributivity (2006, methodology): “This approach was explored in [12] (see also [2]) and involves a technique known as internalization.”
Higher gauge theory (2005, background): “Just as ordinary electromagnetism is described by a connection on a U(1) bundle, usually called the ‘vector potential’ and denoted A, the stringy analogue of electromagnetism is described by a connection on a U(1) gerbe, called the B field.”
Coherent presentations of Artin monoids (2012, background): “Among concrete target 2 -categories for 2 -representations, natural choices are the 2 -categories of 2 -vector spaces, either from Kapranov and Voevodsky [25] or from Baez and Crans [2], of 2 -Hilbert spaces [1] or of categories [13].”
On weak Lie 2‐algebras (2007, methodology): “Theorem 2.1. ([ 1 ]) The 2-functors N and give an equivalence of 2-categories 2Vect and 2Term.”
Membrane sigma-models and quantization of non-geometric flux backgrounds (2012, methodology): “There is a bijective correspondence between semistrict Lie 2-algebras and certain classifying “Postnikov data” [6], analogous to the Faulkner construction of 3-Lie algebras.”
Higher gauge theory I: 2-Bundles (2004, methodology): “Although not directly relevant to this paper, Baez and Alissa Crans [ HDA6 ] have developed a corresponding theory of Lie 2-algebras; here one finds a canonical 1-parameter family of categorifications of a Lie algebra— but these categorifications must be so called ‘weak’ Lie 2-algebras (which corres”
Integrating $L_\infty $-algebras (2006, background): “s (f 0,β 0),(f 1,β 1) ∈( R str) 2 3Not to be confused with the continuous cohomology of G! INTEGRATING L∞-ALGEBRAS 27 are identiﬁed in (τ≤2 R str) 2 if they are simplicially homotopic relatively to ∂∆[2]. This happens if there’s a homotopy F: ∆2 ×[0,1] →Gfrom f 0 to f 1, ﬁxing ∂∆2, and a 2-form Bon ”
The Hom–Yang–Baxter equation, Hom–Lie algebras, and quasi-triangular bialgebras (2009, background): “Also, every Lie algebra L gives a solution of the YBE [ 2 ].”
Twisted Differential String and Fivebrane Structures (2009, background): “• If g is concentrated in degrees 0 through −(n − 1), then we speak of a Lie n-algebra, (for instance the Lie 2-algebras in [8] or the various Lie n-algebras in [68].”
L∞‐Algebras of Classical Field Theories and the Batalin–Vilkovisky Formalism (2018, background): “The categorical equivalence has been proven for Lie 2algebras and 2-term L8-algebras [40]; the extension to Lie n-algebras and n-term L8-algebras should be very involved, but ultimately a mere technicality.”
Representation theory of 2-groups on Kapranov and Voevodsky's 2-vector spaces (2004, background): “…on finite sets both live in Rep2MatC (G), the first as the monoidal category of endomorphisms of the trivial representation (more generally, as the category of morphisms between suitable 1-dimensional representations) and the second as a subcategory of the homotopy category of Rep 2MatC (G).”
Higher-Dimensional Algebra V: 2-Groups (2003, neutral): “ar X iv :m at h/ 03 07”