Bianchi Classification - Classification in Dimension 3

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: Nilpotent and unimodular: Heisenberg algebra. 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: Solvable and not unimodular. 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.) 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: Solvable and not unimodular. = 0, = y, = y + z. 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 semisimple.
• Type V: Solvable and not unimodular. = 0, = y, = z. (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 semisimple.
• Type VI: Solvable and not unimodular. An infinite family. Semidirect products of R2 by R, where the matrix M has non-zero distinct real eigenvalues with non-zero sum. 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: Solvable and unimodular. 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 the Lie algebra of 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: Solvable and not unimodular. An infinite family. Semidirect products of R2 by R, where the matrix M has non-real and non-imaginary eigenvalues. The simply connected group has trivial center and outer automorphism group the non-zero reals.
• Type VII₀: Solvable and unimodular. Semidirect products of R2 by R, where the matrix M has non-zero imaginary eigenvalues. 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: Semisimple and unimodular. The Lie algebra sl₂(R) of traceless 2 by 2 matrices. The simply connected group has center Z and its outer automorphism group has order 2.
• Type IX: Semisimple and unimodular. The Lie algebra of the orthogonal group O₃(R). The simply connected group has center of order 2 and trivial outer automorphism group, and is a spin group.

The classification of 3-dimensional complex Lie algebras is similar except that types VIII and IX become isomorphic, and types VI and VII both become part of a single family of Lie algebras.

The connected 3-dimensional Lie groups can be classified as follows: they are a quotient of the corresponding simply connected Lie group by a discrete subgroup of the center, so can be read off from the table above.

The groups are related to the 8 geometries of Thurston's geometrization conjecture. More precisely, seven of the 8 geometries can be realized as a left-invariant metric on the simply connected group (sometimes in more than one way). The Thurston geometry of type S2×R cannot be realized in this way.