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In mathematics, the Bianchi classification, named for Luigi Bianchi, is a classification of 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 two of the groups are included in the infinite families, giving 9 instead of 11 classes.)

Dimension 0

The only Lie algebra is the abelian Lie algebra R1, with outer automorphism group the group of non-zero real numbers.

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.

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.

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 \( n^{cd};[ \)1] 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.[2] 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; its analysis is referred to as the BKL analysis after Belinskii, Khalatnikov and Lifshitz. [3] [4] 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.[5][6][7] Other more recent work is concerned with the discrete nature of the Kasner map and a continuous generalisation.[8][9][10]
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


Lev Landau and Evgeny Lifshitz (1980), Course of Theoretical Physics vol. 2: The Classical Theory of Fields, Butterworth-Heinemann, ISBN 978-0-7506-2768-9
Robert Wald, General Relativity, University of Chicago Press (1984). ISBN 0-226-87033-2, (chapt 7.2, pages 168–179)
V. A. Belinskii, I. M. Khalatnikov, and E. M. Lifshitz, Zh. Eksp. Teor. Fiz. 62, 1606 (1972)
V. A. Belinskii, I. M. Khalatnikov, and E. M. Lifshitz, Zh. Eksp. Teor. Fiz. 60, 1969 (1971)
M. Henneaux, D. Persson, and P. Spindel, Living Reviews in Relativity 11, 1 (2008), 0710.1818
M. Henneaux, D. Persson, and D. H. Wesley, Journal of High Energy Physics 2008, 052 (2008)
M. Henneaux, ArXiv e-prints (2008), 0806.4670
N. J. Cornish and J. J. Levin, in Recent Developments in Theoretical and Experimental General Relativity, Gravitation, and Relativistic Field Theories, edited by T. Piran and R. Ruffini (1999), pp. 616–+
N. J. Cornish and J. J. Levin, Phys. Rev. Lett. 78, 998 (1997)

N. J. Cornish and J. J. Levin, Phys. Rev. D 55, 7489 (1997)

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

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