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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: December 15, 2011
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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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Additional Information

Theodore Th. Voronov
Affiliation: School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
Email: theodore.voronov@manchester.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11116-X
PII: S 0002-9939(2011)11116-X
Keywords: Maurer–Cartan equation, Lie superalgebras, differential forms, supermanifolds, Lie algebroids, homological vector fields, multiplicative integral, $Q$-manifolds, Quillen’s superconnection
Received by editor(s): December 4, 2009
Received by editor(s) in revised form: March 10, 2011
Published electronically: December 15, 2011
Communicated by: Varghese Mathai
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.