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 "sl"₂(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 "O"₃(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 "S"²"×R cannot be realized in this way.

tructure constants

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

:$left(\; frac\{partial\; xi^\{(c)\}\_i\}\{partial\; x^k\}\; -\; frac\{partial\; xi^\{(c)\}\_k\}\{partial\; x^i\}\; ight)\; xi^i\_\{(a)\}\; xi^k\_\{(b)\}\; =\; C^c\_\{\; ab\}$

where $C^c\_\{\; ab\}$, the "structure constants" of the group, form a constant rank-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 "i"th eigenvalue of $n^\{cd\}$ [cite |title=Course of Theoretical Physics vol. 2: The Classical Theory of Fields |author=Lev Landau and Evgeny Lifshitz |isbn=978-0750627689 |date=1980 |publisher=Butterworth-Heinemann] ; the parameter "a" runs over all positive real numbers:

Cosmological application

In cosmology, this classification is used for a homogeneous spacetime of dimension 3+1. The Friedmann-Lemaître-Robertson-Walker metric is isotropic, which is a particular case of types I, V and IX. A Bianchi type IX cosmology has as special cases the Kasner metric and Taub metric. [Robert Wald, "General Relativity", University of Chicago Press (1984). ISBN 0226870332, (chapt 7.2, pages 168 - 179)]

Curvature of Bianchi spaces

The Bianchi spaces have the property that their Ricci tensors can 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:

:$R\_\{ik\}\; =\; R\_\{(a)(b)\}\; xi^\{(a)\}\_i\; xi^\{(b)\}\_k$

:$R\_\{(a)(b)\}\; =\; frac\{1\}\{2\}\; left\; [\; C^\{cd\}\_\{\; b\}\; left(\; C\_\{cda\}\; +\; C\_\{dca\}\; ight)\; +\; C^c\_\{\; cd\}\; left(\; C^\{\; d\}\_\{ab\}\; +\; C^\{\; d\}\_\{ba\}\; ight)\; -\; frac\{1\}\{2\}\; C^\{\; cd\}\_b\; C\_\{acd\}\; ight]$

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

ee 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) [http://ipsapp007.kluweronline.com/content/getfile/4728/60/13/abstract.htm 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, [http://www34.homepage.villanova.edu/robert.jantzen/bianchi/ Bianchi classification of 3-geometries: original papers in translation]