# 34. The Holonomy Algebra

Let $ \pi \colon E \to M$ be a vector bundle with connection $ \nabla$. We know from Lecture 32 that each holonomy group $ \operatorname{Hol}^{\nabla}(p)$ is a Lie subgroup of $ \operatorname{GL}(E_p)$. Thus its Lie algebra, denoted by $ \mathfrak{hol}^{\nabla}(p)$, is a Lie subalgebra of $ \mathfrak{gl}(E_p)$.

We form the *holonomy algebra*** **of $ \nabla$ as the vector bundle

$$ \mathfrak{hol}^{\nabla} := \bigsqcup_{p \in M} \mathfrak{hol}^{\nabla}(p).$$

In this lecture we prove that the holonomy algebra is a Lie algebra subbundle of the endomorphism bundle $\operatorname{End}(E)$.

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