Bianchi classification

In mathematics, the Bianchi classification provides a list of all real 3-dimensional Lie algebras (up to isomorphism). The classification contains 11 classes, 9 of which contain a single Lie algebra and two of which contain a continuum-sized family of Lie algebras. (Sometimes two of the groups are included in the infinite families, giving 9 instead of 11 classes.) The classification is important in geometry and physics, because the associated Lie groups serve as symmetry groups of 3-dimensional Riemannian manifolds. It is named for Luigi Bianchi, who worked it out in 1898. The term "Bianchi classification" is also used for similar classifications in other dimensions and for classifications of complex Lie algebras. 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 multiplicative 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. It has trivial center and trivial outer automorphism group. The associated simply connected Lie group is the affine group of the line. All the 3-dimensional Lie algebras other than types VIII and IX can be constructed as a semidirect product of R2 and R, with R acting on R2 by some 2 by 2 matrix M. The different types correspond to different types of matrices M, as described below. Type I: This is the abelian and unimodular Lie algebra R3. The simply connected group has center R3 and outer automorphism group GL3(R). This is the case when M is 0. Type II: The Heisenberg algebra, which is nilpotent and unimodular. The simply connected group has center R and outer automorphism group GL2(R). This is the case when M is nilpotent but not 0 (eigenvalues all 0). Type III: This algebra is a product of R and the 2-dimensional non-abelian Lie algebra. (It is a limiting case of type VI, where one eigenvalue becomes zero.