On a non-Abelian Poincaré lemma

Voronov, Theodore

doi:10.1090/S0002-9939-2011-11116-X

On a non-Abelian Poincaré lemma
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by Theodore Th. Voronov PDF

Proc. Amer. Math. Soc. 140 (2012), 2855-2872 Request permission

Abstract:

We show that a well-known result on solutions of the Maurer–Cartan equation extends to arbitrary (inhomogeneous) odd forms: any such form with values in a Lie superalgebra satisfying $d\omega +\omega ^2=0$ is gauge-equivalent to a constant, \[ \omega =gCg^{-1}-dg g^{-1}.\] This follows from a non-Abelian version of a chain homotopy formula making use of multiplicative integrals. An application to Lie algebroids and their non-linear analogs is given.

Constructions presented here generalize to an abstract setting of differential Lie superalgebras where we arrive at the statement that odd elements (not necessarily satisfying the Maurer–Cartan equation) are homotopic — in a certain particular sense — if and only if they are gauge-equivalent.

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