Bianchi classification
In mathematics, the Bianchi classification, named for Luigi Bianchi, is a classification of Lie algebras.
In zero dimensions, the only Lie algebra is the abelian Lie algebra R0. In one dimension, the only Lie algebra is the abelian Lie algebra R1, with outer automorphism group the group of non-zero real numbers.
In two dimensions, there are two Lie algebras:
The system classifies 3-dimensional real Lie algebras into 11 classes, 9 of which are single groups and two of which have a continuum of isomorphism classes. Sometimes type Type VI and Type VII groups are included in the infinite families, giving 9 instead of 11 classes.
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.
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.
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).
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.
[y,z] = 0, [x,y] = y, [x, z] = 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.
[y,z] = 0, [x,y] = y, [x, z] = 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.
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.
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