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I have seen two kinds of definitions of irreducible connections on fibre bundls: A connection is said to be irreducible if

  1. the holonomy group is precisely $G$ and not a proper subgroup.

or 2. there exist a constant $C$ depending on the connection $D$ such that \begin{equation} \|s\|_{H^{1,2}} \le C\|Ds\|_{L^2} \end{equation} for all $s\in H^{1,2}(\Omega^0(ad\eta))$ .

My question is: what is the relation and difference of the above two definitions?

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For a connection A, Let $\Gamma_A=\{u\in \mathscr{G}\ |\ u(A)=A\}$, $\mathscr{G}$ is the gauge group. Let $H_A$ be the holonomy group correspond to connection A, then $\Gamma_A$ is isomorphic to the centralizer of $H_A$ in G.

For $u\in\ \Gamma_A$, $u(A)=u-(\nabla_Au)u^{-1}$, so the lie algebra of $\Gamma_A$ correspond to kernel of $\nabla_A$, that is your second definition. The inverse is similar.

You could find above related topic in Geometry of 4-manifold, S.K.Donaldson P.B.Kronheimer in pages near Lemma (4.2.8)

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