strict Lie 2-algebra
Formal Lie groupoids
A strict Lie 2-algebra is the infinitesimal approximation to a smooth strict 2-group in generalization of how an ordinary Lie algebra is the infinitesimal approximation to a Lie group.
The notion of strict Lie 2-algebra is the special case of a general notion Lie 2-algebra for which the Jacobi identity does hold (and not just up to nontrivial isomorphism).
More precisely: a strict Lie 2-algebra is an ∞-Lie algebra with generators just in degree 1 and 2 and at most the unary and binary brackets being nontrivial.
Equivalently, this is a dg-Lie algebra with generators in the lowest two degrees.
In direct analogy to how strict 2-groups are equivalently encoded in a smooth crossed module of groups, a strict Lie 2-algebra is equivalently encoded in a differential crossed module of ordinary Lie algebras.
Derivation Lie 2-algebra
The Lie version of a smooth automorphism 2-group is the derivation Lie 2-algebra of an ordinary Lie algebra . This is the one coming from the differential crossed module .
Revised on August 28, 2011 12:57:51
by Urs Schreiber