nLab
Cartan's homotopy formula

Let MM be a differentiable manifold, XX a vector field on MM, and X\mathcal{L}_X the Lie derivative along XX. The contraction of a vector field and a kk-form ω\omega is denoted in modern literature by XωX \rfloor \omega (should be lrcorner instead of rfloor LaTeX command, but it does not work in iTeX) or ι Xω\iota_X \omega or ι(X)(ω)\iota(X)(\omega).

Then the Cartan’s infinitesimal homotopy formula, nowdays called simply Cartan’s homotopy formula or even Cartan formula, says

Xω=dι(X)ω+ι(X)dω \mathcal{L}_X \omega = d \iota(X)\omega + \iota(X) d\omega

The word “homotopy” is used because it supplies a homotopy operator for some manipulation with chain complexes in de Rham cohomology. Cartan’s homotopy formula is part of Cartan calculus.

Regarding that the Cartan’s formula can be viewed as a formula about the de Rham complex, which has generalizations, one can often define the Cartan’s formulas for those generalizations. For example

  • Masoud Khalkhali, On Cartan homotopy formulas in cyclic homology, Manuscripta mathematica 94:1, pp 111-132 (1997) doi

See also noncommutative differential calculus where it is incorporated into the notion of Batalin-Vilkovisky module over a Gerstenhaber algebra.

Revised on October 14, 2014 17:08:52 by Zoran Škoda (161.53.130.104)