Bianchi classification

From Wikipedia, the free encyclopedia

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 $\xi _{i}^{{(a)}}$ which obey the following property:

$\left({\frac {\partial \xi _{i}^{{(c)}}}{\partial x^{k}}}-{\frac {\partial \xi _{k}^{{(c)}}}{\partial x^{i}}}\right)\xi _{{(a)}}^{i}\xi _{{(b)}}^{k}=C_{{\ ab}}^{c}$

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

$C_{{\ ab}}^{c}=\varepsilon _{{abd}}n^{{cd}}-\delta _{a}^{c}a_{b}+\delta _{b}^{c}a_{a}$

where $\varepsilon _{{abd}}$ is the Levi-Civita symbol, $\delta _{a}^{c}$ 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 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

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, and 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

$ds^{2}=\gamma _{{ab}}\xi _{i}^{{(a)}}\xi _{k}^{{(b)}}dx^{i}dx^{k}$ (where $\xi _{i}^{{(a)}}dx^{i}$

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

$R_{{ik}}=R_{{(a)(b)}}\xi _{i}^{{(a)}}\xi _{k}^{{(b)}}$

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

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

References

↑ 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

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.