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In the definition of homogenous vector bundles, an equivalence class is defined.


G is a lie group and H a (lie) subgroup. Define

$$ \rho : H \rightarrow GL(V) $$

where V is a vector space.

The equivalence class is defined as

$$ (g_1, v_1) = (g_1h, \rho(h^{-1})v_1) $$

The question is why the above definition and not

$$ (g_1, v_1) = (g_1h, \rho(h)v_1) ? $$

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The first definition $$ (g, v).h ~:=~ (gh, \rho(h^{-1})v) $$ defines a right group action $G \times V \times H \to G \times V$, $$ ((g, v).h).k~=~(g, v).(h.k), $$ while the second definition $$ (g, v) ~\mapsto~ (gh, \rho(h)v) $$ is neither a left nor a right group action, cf. Chris Gerig's comment.

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This completes the picture. The second definition wouldn't work in an obvious way – Tbh Dec 20 '11 at 18:21

Following up on Qmechanic's response, here is a a step by step demonstration of why one definition works and the other does not:

$$ (g,v)hk = ((g,v)h)k = ((gh, \rho(h)v)k = ((ghk, \rho(k)\rho(h)v) = ((ghk, \rho(kh)v) \neq (g,v)hk $$

From the above the second definition does not work.

On the other hand

$$ (g,v)hk = ((g,v)h)k = ((gh, \rho(h^{-1})v)k = ((ghk, \rho(k^{-1})\rho(h^{-1})v) = ((ghk, \rho(k^{-1}h^{-1})v) = $$

$$ ((ghk, \rho((hk)^{-1})v) = (g,v)hk $$

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