Abstract
Hyperbolic algebras such as E10 are based on Minkowskian root spaces that revert to finite or affine root systems upon the removal of any simple root. In string theory terms, finite Lie, affine Kac-Moody and hyperbolic Kac-Moody algebras are generated by tachyon vertex operators, trachyon plus photon vertex operators, and vertex operators for all mass levels, respectively. The 136 possible hyperbolic Dynkin diagrams between the ranks 3 and 10 are classified and exhibited, completing an earlier enumeration by Kac of the 18 rank 7-10 cases. The rank-2 hyperbolic algebras, infinite in number, have been classified by Lepowsky and Moody (1979).
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