Non-abelian extensions of Lie 2-algebras

Abstract

In this paper, we introduce the notion of derivations of Lie 2-algebras and construct the associated derivation Lie 3-algebra. We prove that isomorphism classes of non-abelian extensions of Lie 2-algebras are classified by equivalence classes of morphisms from a Lie 2-algebra to a derivation Lie 3-algebra.

Crossed Modules for Lie 2-Algebras

“…Let m = g, i = Id in Example 3.9. Then Theorem 4.4 recovers the theorem that Der(g) is a strict Lie 2-algebra in [5] and Proposition 4.5 implies that H 1 (g) = Der 0 (g)/Inn 0 (g) is a quotient Lie algebra. Also, it justifies our definition of Inn 0 (g), while it is defined by Im(D| g0 ) in [5] which is not an ideal of Der 0 (g) by (24).…”

Cohomological characterizations of non-abelian extensions of strict Lie 2-algebras

“…We construct a strict Lie 2-algebra SDer(g) associated to strict derivations, which plays important role when we consider non-abelian extensions of strict Lie 2-algebras. This part is not totally new, and one can obtain these results from [7,8]. Then we show that a non-abelian extension of a strict Lie 2algebra g by h naturally gives a strict homomorphism from g to SOut(h).…”

“…Some work on the study of non-abelian extensions of Lie 2-algebras has been done recently. In [7], the authors showed that a non-abelian extension of a Lie 2-algebra g by a Lie 2-algebra h can be characterized by a Lie 3-algebra homomorphism from g to the derivation Lie 3-algebra DER(h). In [19], the author classified general non-abelian extensions of L ∞ -algebras via homotopy classes of L ∞ -morphism.…”

Integration of Derivations for Lie 2-Algebras

“…Let (Der(g), {·, ·}) be the derivation Lie 2-algebra of g, and denote the corresponding crossed module of Lie algebras by (Der −1 (g), Der 0 (g),d, φ), where the action φ is given by (5), and the Lie bracket on Der −1 (g) is given by…”

“…To study nonabelian extensions of Lie 2-algebras, the authors introduced the notion of a derivation of a Lie 2-algebra g = (g 0 ⊕g −1 , d, [·, ·], l 3 ) in [5], and showed that there is a strict Lie 2-algebra Der(g), in which the degree −1 part is the set of degree −1 derivations Der −1 (g), and the degree 0 part is the set of degree 0 derivations Der 0 (g). Furthermore, one can also construct a Lie 3-algebra DER(g) (called the derivation Lie 3-algebra), in which the degree −2 part is g −1 , the degree −1 part is Der −1 (g) ⊕ g 0 , and the degree 0 part is Der 0 (g).…”