Bianchi Classification - Classification in Dimension Less Than 3

Classification in Dimension Less Than 3

  • Dimension 0: The only Lie algebra is the abelian Lie algebra R0.
  • Dimension 1: The only Lie algebra is the abelian Lie algebra R1, with outer automorphism group the group of non-zero real numbers.
  • Dimension 2: There are two Lie algebras:
(1) The abelian Lie algebra R2, with outer automorphism group GL2(R).
(2) The solvable Lie algebra of 2×2 upper triangular matrices of trace 0. The simply connected group has trivial center and outer automorphism group of order 2.

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