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.

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 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.

Classification in dimension 3[]

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.) It is solvable and not unimodular. The simply connected group has center R and outer automorphism group the group of non-zero real numbers. The matrix M has one zero and one non-zero eigenvalue.
• Type IV: The algebra generated by [y,z] = 0, [x,y] = y, [x, z] = y + z. It is solvable and not unimodular. The simply connected group has trivial center and outer automorphism group the product of the reals and a group of order 2. The matrix M has two equal non-zero eigenvalues, but is not diagonalizable.
• Type V: [y,z] = 0, [x,y] = y, [x, z] = z. Solvable and not unimodular. (A limiting case of type VI where both eigenvalues are equal.) The simply connected group has trivial center and outer automorphism group the elements of GL₂(R) of determinant +1 or −1. The matrix M has two equal eigenvalues, and is diagonalizable.
• Type VI: An infinite family: semidirect products of R² by R, where the matrix M has non-zero distinct real eigenvalues with non-zero sum. The algebras are solvable and not unimodular. The simply connected group has trivial center and outer automorphism group a product of the non-zero real numbers and a group of order 2.
• Type VI₀: This Lie algebra is the semidirect product of R² by R, with R where the matrix M has non-zero distinct real eigenvalues with zero sum. It is solvable and unimodular. It is the Lie algebra of the 2-dimensional Poincaré group, the group of isometries of 2-dimensional Minkowski space. The simply connected group has trivial center and outer automorphism group the product of the positive real numbers with the dihedral group of order 8.
• Type VII: An infinite family: semidirect products of R² by R, where the matrix M has non-real and non-imaginary eigenvalues. Solvable and not unimodular. The simply connected group has trivial center and outer automorphism group the non-zero reals.
• Type VII₀: Semidirect product of R² by R, where the matrix M has non-zero imaginary eigenvalues. Solvable and unimodular. This is the Lie algebra of the group of isometries of the plane. The simply connected group has center Z and outer automorphism group a product of the non-zero real numbers and a group of order 2.
• Type VIII: The Lie algebra sl₂(R) of traceless 2 by 2 matrices, associated to the group SL₂(R). It is simple and unimodular. The simply connected group is not a matrix group; it is denoted by ${\displaystyle {\overline {{\mbox{SL}}(2,\mathbf {R} )}}}$, has center Z and its outer automorphism group has order 2.
• Type IX: The Lie algebra of the orthogonal group O₃(R). It is denoted by