Uncovering the symmetries on $[p,q]$ 7-branes: Beyond the Kodaira classification

We begin a classification of the symmetry algebras arising on configurations of type IIB [p, q] 7-branes. These include not just the Kodaira symmetries that occur when branes coalesce into a singularity, but also algebras associated to other physically interesting brane configurations that cannot be collapsed.