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Covariant derivatives

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I need correctly define covariant derivatives on the coset space $G/H$, where a group $G \equiv \{X_i, Y_a\}$ ($X$ and $Y$ are generators) have a subrgroup $H \equiv \{X_i\}$ Lie algebra of $G$ has the following structure: $$[X, X] \sim X, \quad [X, Y] \sim Y, \quad [Y, Y] \sim X + Y.$$ The coset space $G/H$ is parametrized by coordinates $\xi^a$ corresponding to the generators $Y_a$. In the exponential parametrization an element of this coset has the form $$G/H: \quad \Omega = exp\{i\xi^a Y_a\}$$ The ordinary derivatives $\frac{\partial}{\partial\xi^a}$ of $\psi(\xi)$ ($\psi(\xi)$ - field defined on $G/H$) do not transform covariantly.

I was told once that it is quite easy to define covariant derivatives using the Cartan form $\Omega^{-1}d\Omega$.

I looked through the bunch of the mathematical books about Lie groups and Lie algebras but I can't find a clear description of the procedure. Can anyone advise me a proper book (paper) or, if it's not take much time, tell it here?

Thank you.

Sorry for my English)


This post has been migrated from (A51.SE)

asked Oct 8, 2011 in Mathematics by Vladimir Bychkov (45 points) [ revision history ]
recategorized Dec 31, 2015 by Dilaton
A clear exposition of the geometry of homogeneous spaces can be found in Appendix A of the review article: Harmonic analysis and propagators on homogeneous spaces by Roberto Camporesi Phys. Rep. 196 (1990) 1-134.the article can be downloaded from Roberto Camporesi's homepage: http://calvino.polito.it/~camporesi/.

This post has been migrated from (A51.SE)

1 Answer

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There are a number of imprecisions in your question, mostly having to do with confusing the Lie group and its Lie algebra. I suppose this will make it hard to read the mathematical literature. Having said that, the first volume of Kobayashi and Nomizu is probably the canonical reference.

Let me try to summarise. Let me assume that $H$ is connected.

The structure of the split $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$ of the Lie algebra $\mathfrak{g}$ of $G$ into the Lie algebra $\mathfrak{h}$ of $H$ and the complement $\mathfrak{m} = \mathrm{Span}(\lbrace Y_a\rbrace)$, says that you have a reductive homogeneous space. Such homogeneous spaces have a canonical invariant connection and hence a canonical notion of covariant derivative.

The map $G \to G/H$ defines a principal $H$-bundle. Your $\Omega$ is a local section of this bundle. On $G$ you have the left-invariant Maurer-Cartan one-form $\Theta$, which is $\mathfrak{g}$-valued. You can use $\Omega$ to pull back $\Theta$ to $G/H$: it is a locally defined one-form on $G/H$ with values in $\mathfrak{g}$. For matrix groups, it is indeed the case that $\Omega^*\Theta = \Omega^{-1}d\Omega$, but you can in fact use this notation for most computations without worrying too much.

Decompose $\Omega^{-1}d\Omega$ according to the split $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$: $$ \Omega^{-1} d\Omega = \omega + \theta $$ where $\omega$ is the $\mathfrak{h}$ component and $\theta$ is the $\mathfrak{m}$ component. It follows that $\theta$ defines pointwise an $H$-equivariant isomorphism from the tangent space to $G/H$ and $\mathfrak{m}$, with $H$ acting on $\mathfrak{m}$ by the restriction to $H$ of the adjoint action of $G$ on $\mathfrak{g}$ and $H$ acting on $G/H$ via the linear isotropy representation. This means that $\theta$ is a soldering form.

On the other hand $\omega$ is $\mathfrak{h}$-valued and defines a connection one-form. You can check that if you change the parametrisation $\Omega$, then $\omega$ does transform as a connection under the local $H$-transformations.

This then allows you to differentiate sections of homogeneous vector bundles on $G/H$, such as tensors. In your notation and assuming that $\psi$ is a section of one such bundle, associated to a representation $\rho$ of $H$, the covariant derivative would be $$ \nabla \psi = d\psi + \rho(\omega)\psi~, $$ where I also denote by $\rho$ the representation of the Lie algebra of $H$.

The Maurer-Cartan structure equation satisfied by $\Theta$ is $$ d\Theta = - \tfrac12 [\Theta,\Theta] $$ and this pulls back to $G/H$ to give the following equations $$ d\theta + [\omega,\theta] = -\tfrac12 [\theta,\theta]_{\mathfrak{m}} $$ and $$ d\omega + \tfrac12 [\omega,\omega] = - \tfrac12 [\theta,\theta]_{\mathfrak{h}} $$ which say that the torsion $T$ and curvature $K$ of $\omega$ are given respectively by $$ T = -\tfrac12 [\theta,\theta]_{\mathfrak{m}} \qquad\mathrm{and}\qquad K = - \tfrac12 [\theta,\theta]_{\mathfrak{h}}. $$

One thing to keep in mind is that in general $\nabla$ will not be the Levi-Civita connection of any invariant metric, since it has torsion. (If (and only if) the torsion vanishes, you have a (locally) symmetric space.) If you are interested in the Levi-Civita connection of an invariant metric, then you have to modify the invariant connection by the addition of a contorsion tensor which kills the torsion. The details are not hard to work out.

This post has been migrated from (A51.SE)
answered Oct 8, 2011 by José Figueroa-O'Farrill (2,125 points) [ no revision ]

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