Bianchi classification

In mathematics, the Bianchi classification, named for Luigi Bianchi, is a classification of the 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 two of the groups are included in the infinite families, giving 9 instead of 11 classes.) The term "Bianchi classification" is also used for similar classifications in other dimensions.

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.

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. [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.
• Type V: Solvable and not unimodular. [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.
• 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.

Structure constants

The three-dimensional Bianchi spaces each admit a set of three Killing vectors $\xi^{(a)}_i$ which obey the following property:

$\left( \frac{\partial \xi^{(c)}_i}{\partial x^k} - \frac{\partial \xi^{(c)}_k}{\partial x^i} \right) \xi^i_{(a)} \xi^k_{(b)} = C^c_{\ ab}$

where $C^c_{\ ab}$, the "structure constants" of the group, form a constant order-three tensor antisymmetric in its lower two indices. For any three-dimensional Bianchi space, $C^c_{\ ab}$ is given by the relationship

$C^c_{\ ab} = \varepsilon_{abd}n^{cd} - \delta^c_a a_b + \delta^c_b a_a$

where $\varepsilon_{abd}$ is the Levi-Civita symbol, $\delta^c_a$ is the Kronecker delta, and the vector $a_a = (a,0,0)$ and diagonal tensor $n^{cd}$ are described by the following table, where $n^{(i)}$ gives the ith eigenvalue of $n^{cd}$; the parameter a runs over all positive real numbers:

Bianchi type	$a$	$n^{(1)}$	$n^{(2)}$	$n^{(3)}$	notes
I	0	0	0	0	describes Euclidean space
II	0	1	0	0
III	1	0	1	-1	the subcase of type VIa with $a = 1$
IV	1	0	0	1
V	1	0	0	0	has a hyper-pseudosphere as a special case
VI0	0	1	-1	0
VIa	$a$	0	1	-1	when $a = 1$, equivalent to type III
VII0	0	1	1	0	has Euclidean space as a special case
VIIa	$a$	0	1	1	has a hyper-pseudosphere as a special case
VIII	0	1	1	-1
IX	0	1	1	1	has a hypersphere as a special case

Cosmological application

In cosmology, this classification is used for a homogeneous spacetime of dimension 3+1. The Friedmann-Lemaître-Robertson-Walker metrics are isotropic, which are particular cases of types I, V, $\scriptstyle\text{VII}_h$ and IX. The Bianchi type I models include the Kasner metric as a special case. The Bianchi IX cosmologies include the Taub metric. However, the dynamics near the singularity is approximately governed by a series of successive Kasner (Bianchi I) periods. The complicated dynamics, which essentially amounts to billiard motion in a portion of hyperbolic space, exhibits chaotic behaviour, and is named Mixmaster, and its analysis is referred to as the BKL analysis after Belinskii, Khalatnikov and Lifshitz. More recent work has established a relation of (super-)gravity theories near a spacelike singularity (BKL-limit) with Lorentzian Kac-Moody algebras, Weyl groups and hyperbolic Coxeter groups. Other more recent work is concerned with the discrete nature of the Kasner map and a continuous generalisation.

Curvature of Bianchi spaces

The Bianchi spaces have the property that their Ricci tensors can be separated into a product of the basis vectors associated with the space and a coordinate-independent tensor.

For a given metric

$ds^2 = \gamma_{ab} \xi^{(a)}_i \xi^{(b)}_k dx^i dx^k$ (where $\xi^{(a)}_idx^i$

are 1-forms), the Ricci curvature tensor $R_{ik}$ is given by:

$R_{ik} = R_{(a)(b)} \xi^{(a)}_i \xi^{(b)}_k$

$R_{(a)(b)} = \frac{1}{2} \left[ C^{cd}_{\ \ b} \left( C_{cda} + C_{dca} \right) + C^c_{\ cd} \left( C^{\ \ d}_{ab} + C^{\ \ d}_{ba} \right) - \frac{1}{2} C^{\ cd}_b C_{acd} \right]$

where the indices on the structure constants are raised and lowered with $\gamma_{ab}$ which is not a function of $x^i$.

See also

• Table of Lie groups
• List of simple Lie groups

References

• L. Bianchi, Sugli spazii a tre dimensioni che ammettono un gruppo continuo di movimenti. (On the spaces of three dimensions that admit a continuous group of movements.) Soc. Ital. Sci. Mem. di Mat. 11, 267 (1898) English translation
• Guido Fubini Sugli spazi a quattro dimensioni che ammettono un gruppo continuo di movimenti, (On the spaces of four dimensions that admit a continuous group of movements.) Ann. Mat. pura appli. (3) 9, 33-90 (1904); reprinted in Opere Scelte, a cura dell'Unione matematica italiana e col contributo del Consiglio nazionale delle ricerche, Roma Edizioni Cremonese, 1957-62
• MacCallum, On the classification of the real four-dimensional Lie algebras, in "On Einstein's path: essays in honor of Engelbert Schucking" edited by A. L. Harvey , Springer ISBN 0-387-98564-6
• Robert T. Jantzen, Bianchi classification of 3-geometries: original papers in translation