In Penrose abstract index notation, the expression "raising and lowering indices" refers to the following.
Suppose we have a Riemannian metric $g^{ab}$. We want that, given $\alpha_b$, we can simply write $\alpha^a$ to denote $g^{ab}\alpha_b$ and, reciprocally, given $\alpha^b$, $\alpha_a$ to denote $g_{ab}\alpha^b$. But we can have problems with tensors with both kind of indices (upper and lower). For example, to denote
$$ g^{mz}\alpha_{xyz}^{abc} $$what will we choose from $\alpha_{xy}^{abcm}$, $\alpha_{xy}^{abmc}$, $\alpha_{xy}^{ambc}$, $\alpha_{xy}^{mabc}$?
We could take as a convention that we will always put the new index the first one ($\alpha_{xy}^{mabc}$).
But this is not consistent because if we want to lower again we don't recover the original tensor. For example:
$$ \alpha_{xyw}^{abc}=\delta_w^z\alpha_{xyz}^{abc}=g_{wm}g^{mz}\alpha_{xyz}^{abc}=g_{wm}\alpha_{xy}^{mabc}=\alpha_{wxy}^{abc} $$and this is not true in general.
For this reason, from now on we will denote tensors with some slots to indicate the order:
$$ \beta_{x\hspace{0.15cm} y}^{\hspace{0.15cm} a} $$Also, it can be shown easily that $\delta _ {\hspace{0.15cm} b } ^ { a } = g _ {\hspace{0.15cm} b } ^ { a }$ and $\delta _ { a b } = g _ { a b }$.
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Author of the notes: Antonio J. Pan-Collantes
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