Homotopy derivations

Abstract

We define a strong homotopy derivation of (cohomological) degree k of a strong homotopy algebra over an operad P. This involves resolving the operad obtained from P by adding a generator with "derivation relations". For a wide class of Koszul operads P, in particular Ass and Lie, we describe the strong homotopy derivations by coderivations and show that they are closed under the Lie bracket. We show that symmetrization of a strong homotopy derivation of an A ∞ algebra yields a strong homotopy derivation of the symmetrized L ∞ algebra. We give examples of strong homotopy derivations generalizing inner derivations.

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).…”

Integration of Derivations for Lie 2-Algebras

“…More explicitly, even for a strict Lie 2-algebra, a homotopy derivation (X, l X ) of degree 0 can have a nonzero homotopy term "l X ", while the strict derivation of degree 0 requires that l X = 0. Moreover, a homotopy derivation of degree −1 is more restrictive than a strict derivation of degree −1 since the former requires that both (7) and (8) hold, and the latter only requires that (8) holds. Our derivation Lie 2-algebra can be viewed as a unification of them in the sense that the Lie 2-algebra of homotopy derivations and the Lie 2-algebra of strict derivations are both sub-Lie 2-algebras of ours.…”

“…Remark 3.10. Parallel to Remark 3.3, we compare our inner derivations with homotopy inner derivations given in [7]. For a semistrict Lie 2-algebra, a homotopy inner derivation of degree 0 is given by (ad 0 (x), l 3 (x, ·, ·)) for x ∈ g 0 , and a homotopy inner derivation of degree −1 is given by ad 1 (a) for a ∈ g −1 satisfying d(a) = 0.…”

“…For a semistrict Lie 2-algebra, a homotopy inner derivation of degree 0 is given by (ad 0 (x), l 3 (x, ·, ·)) for x ∈ g 0 , and a homotopy inner derivation of degree −1 is given by ad 1 (a) for a ∈ g −1 satisfying d(a) = 0. See [7,Proposition 4.4] for general formulas. The Lie 2-algebra of homotopy inner derivations is a sub-Lie 2-algebra of inn(g).…”

“…In Section 3, we work out the derivation Lie 2-algebra for the string Lie 2-algebra. We also compare our derivations with homotopy derivations introduced in [7] recently. In Section 4, we construct a crossed module of Lie groups (Aut −1 (g), Aut 0 (g), ∂, ⊲) associated to automorphisms of a Lie 2-algebra g (Theorem 4.4).…”