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ISSN 1088-6826(e) ISSN 0002-9939(p)

On a non-Abelian Poincaré lemma

Author: Theodore Th. Voronov
Journal: Proc. Amer. Math. Soc. 140 (2012), 2855-2872
MSC (2010): Primary 53D17; Secondary 18G50, 58A50, 58C50, 58H05
Posted: December 15, 2011
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Abstract | References | Similar Articles | Additional Information

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,

$\displaystyle \omega =gCg^{-1}-dg\,g^{-1}.$

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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