  In a previous paper we explored how conjugacy classes of the modular group
classify the symmetry algebras that arise on type IIB [p,q] 7-branes. The
Kodaira list of finite Lie algebras completely fills the elliptic classes as
well as some parabolic classes. Loop algebras of E_N fill additional parabolic
classes, and exotic finite algebras, hyperbolic extensions of E_N and more
general indefinite Lie algebras fill the hyperbolic classes. Since they
correspond to brane configurations that cannot be made into strict
singularities, these non-Kodaira algebras are spectrum generating and organize
towers of massive BPS states into representations. The smallest brane
configuration with unit monodromy gives rise to the loop algebra \hat{E}_9
which plays a central role in the theory. We elucidate the patterns of
enhancement relating E_8, E_9, \hat{E}_9 and E_10. We examine configurations of
24 7-branes relevant to type IIB compactifications on a two-sphere, or F-theory
on K3. A particularly symmetric configuration separates the 7-branes into two
groups of twelve branes and the massive BPS spectrum is organized by E_10 +
E_10.
