BC3, (BC3, C3), (C, BC3), (B3, B), (C, C3). Rank 4: A4, B4, B, C4, C, BC4, (BC4, C4), (C, BC4), (B4, B), (C, C4), D4, F4, F. Rank 5: A5, B5, B, C5, C, BC5, (BC5, C5), (C, BC5), (B5, B), (C, C5), D5. Rank 6: A6, B6, B, C6, C, BC6, (BC6, C6), (C, BC6), (B6, B), (C, C6), D6, E6, Rank 7: A7, B7, B, C7, C, BC7, (BC7, C7), (C, BC7), (B7, B), (C, C7), D7, E7, Rank 8: A8, B8, B, C8, C, BC8, (BC8, C8), (C, BC8), (B8, B), (C, C8), D8, E8, Rank n (n>8): An, Bn, B, Cn, C, BCn, (BCn, Cn), (C, BCn), (Bn, B), (C, Cn), Dn. Applications showed that the affine root systems index Macdonald identities used affine root systems to study p-adic algebraic groups. Reduced affine root systems classify affine Kac–Moody algebras, while the non-reduced affine root systems correspond to affine Lie superalgebras. showed that affine roots systems index families of Macdonald polynomials. References Discrete groups Lie algebras Orthogonal polynomials