Bianchi classification

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.)

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.^[1] 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. ^[2] ^[3] 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.^[4]^[5]^[6] Other more recent work is concerned with the discrete nature of the Kasner map and a continuous generalisation.^[7]^[8]^[9]

Classification

Lower Dimensions

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 abelian Lie algebra R², with outer automorphism group GL₂(R).
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; it is the affine group of the line.

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 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: 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 $n^{cd}$ ;^[10] 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 VI_a 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
VI_a	$a$	0	1	-1	when $a=1$ , equivalent to type III
VII₀	0	1	1	0	has Euclidean space as a special case
VII_a	$a$	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

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 $x^{i}$ .

References

↑ 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)
↑ 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

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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