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| In mathematics, the '''Bianchi classification''', named for [[Luigi Bianchi]], is a classification of the 3-dimensional real [[Lie algebra]]s 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.
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| ==Classification in dimension less than 3==
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| *Dimension 0: The only Lie algebra is the [[abelian Lie algebra]] '''R'''<sup>0</sup>.
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| *Dimension 1: The only Lie algebra is the abelian Lie algebra '''R'''<sup>1</sup>, with outer automorphism group the group of non-zero real numbers.
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| *Dimension 2: There are two Lie algebras:
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| ::(1) The abelian Lie algebra '''R'''<sup>2</sup>, with outer automorphism group GL<sub>2</sub>('''R''').
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| ::(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.
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| ==Classification in dimension 3==
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| All the 3-dimensional Lie algebras other than types VIII and IX can be constructed as a semidirect product of '''R'''<sup>2</sup> and '''R''', with '''R''' acting on '''R'''<sup>2</sup> by some 2 by 2 matrix ''M''. The different types correspond to different types of matrices ''M'', as described below.
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| *'''Type I''': This is the abelian and unimodular Lie algebra '''R'''<sup>3</sup>. The simply connected group has center '''R'''<sup>3</sup> and outer automorphism group GL<sub>3</sub>('''R'''). This is the case when ''M'' is 0.
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| *'''Type II''': Nilpotent and unimodular: [[Heisenberg algebra]]. The simply connected group has center '''R''' and outer automorphism group GL<sub>2</sub>('''R'''). This is the case when ''M'' is nilpotent but not 0 (eigenvalues all 0).
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| *'''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.
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| *'''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.
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| *'''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 GL<sub>2</sub>('''R''') of determinant +1 or −1. The matrix ''M'' has two equal eigenvalues, and is semisimple.
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| *'''Type VI''': Solvable and not unimodular. An infinite family. Semidirect products of '''R'''<sup>2</sup> 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.
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| *'''Type VI<sub>0</sub>''': Solvable and unimodular. This Lie algebra is the semidirect product of '''R'''<sup>2</sup> 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.
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| *'''Type VII''': Solvable and not unimodular. An infinite family. Semidirect products of '''R'''<sup>2</sup> 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.
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| *'''Type VII<sub>0</sub>''': Solvable and unimodular. Semidirect products of '''R'''<sup>2</sup> 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.
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| *'''Type VIII''': Semisimple and unimodular. The Lie algebra ''sl''<sub>2</sub>('''R''') of traceless 2 by 2 matrices. The simply connected group has center '''Z''' and its outer automorphism group has order 2.
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| *'''Type IX''': Semisimple and unimodular. The Lie algebra of the orthogonal group ''O''<sub>3</sub>('''R'''). The simply connected group has center of order 2 and trivial outer automorphism group, and is a [[spin group]].
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| 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.
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| 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.
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| 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''<sup>2</sup>''×'''R''' cannot be realized in this way.
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| ==Structure constants==
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| The three-dimensional Bianchi spaces each admit a set of three [[Killing vector]]s <math>\xi^{(a)}_i</math> which obey the following property:
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| :<math>\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}</math>
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| where <math>C^c_{\ ab}</math>, the "structure constants" of the group, form a [[constant (mathematics)|constant]] [[tensor|order-three tensor]] [[antisymmetric tensor|antisymmetric]] in its lower two indices. For any three-dimensional Bianchi space, <math>C^c_{\ ab}</math> is given by the relationship
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| :<math>C^c_{\ ab} = \varepsilon_{abd}n^{cd} - \delta^c_a a_b + \delta^c_b a_a</math>
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| where <math>\varepsilon_{abd}</math> is the [[Levi-Civita symbol]], <math>\delta^c_a</math> is the [[Kronecker delta]], and the vector <math>a_a = (a,0,0)</math> and [[diagonal matrix|diagonal]] tensor <math>n^{cd}</math> are described by the following table, where <math>n^{(i)}</math> gives the ''i''th [[eigenvalue]] of <math>n^{cd}</math>;<ref>{{citation |title=Course of Theoretical Physics vol. 2: The Classical Theory of Fields |author=[[Lev Landau]] and [[Evgeny Lifshitz]] |isbn=978-0-7506-2768-9 |year=1980 |publisher=Butterworth-Heinemann}}</ref> the parameter ''a'' runs over all positive [[real number]]s:
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| {| class="wikitable" align="center"
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| |-
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| ! Bianchi type
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| ! <math>a</math>
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| ! <math>n^{(1)}</math>
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| ! <math>n^{(2)}</math>
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| ! <math>n^{(3)}</math>
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| ! notes
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| |-
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| | I || 0 || 0 || 0 || 0 || describes [[Euclidean geometry|Euclidean space]]
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| |-
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| | II || 0 || 1 || 0 || 0 ||
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| |-
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| | III || 1 || 0 || 1 || -1 || the subcase of type VI<sub>''a''</sub> with <math>a = 1</math>
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| |-
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| | IV || 1 || 0 || 0 || 1 ||
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| |-
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| | V || 1 || 0 || 0 || 0 || has a hyper-[[pseudosphere]] as a special case
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| |-
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| | VI<sub>0</sub> || 0 || 1 || -1 || 0 ||
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| |-
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| | VI<sub>''a''</sub> || <math>a</math> || 0 || 1 || -1 || when <math>a = 1</math>, equivalent to type III
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| |-
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| | VII<sub>0</sub> || 0 || 1 || 1 || 0 || has Euclidean space as a special case
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| |-
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| | VII<sub>''a''</sub> || <math>a</math> || 0 || 1 || 1 || has a hyper-pseudosphere as a special case
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| |-
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| | VIII || 0 || 1 || 1 || -1 ||
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| |-
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| | IX || 0 || 1 || 1 || 1 || has a [[hypersphere]] as a special case
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| |}
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| == Cosmological application ==
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| In [[cosmology]], this classification is used for a [[homogeneous space|homogeneous]] [[spacetime]] of dimension 3+1. The [[Friedmann-Lemaître-Robertson-Walker metric]]s are isotropic, which are particular cases of types I, V, <math>\scriptstyle\text{VII}_h</math> and IX. The Bianchi type I models include the [[Kasner metric]] as a special case.
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| The Bianchi IX cosmologies include the [[Taub-NUT vacuum|Taub metric]].<ref>[[Robert Wald]], ''General Relativity'', [[University of Chicago Press]] (1984). ISBN 0-226-87033-2, (chapt 7.2, pages 168 - 179)</ref> However, the dynamics near the singularity is approximately governed by a series of successive Kasner (Bianchi I) periods. The complicated dynamics,
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| which essentially amounts to billiard motion in a portion of hyperbolic space, exhibits chaotic behaviour, and is named [[Mixmaster universe|Mixmaster]], and its analysis is referred to as the
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| [[BKL singularity|BKL analysis]] after Belinskii, Khalatnikov and Lifshitz.
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| <ref>V. A. Belinskii, I. M. Khalatnikov, and E. M. Lifshitz, Zh. Eksp. Teor. Fiz. 62, 1606 (1972)</ref>
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| <ref>V. A. Belinskii, I. M. Khalatnikov, and E. M. Lifshitz, Zh. Eksp. Teor. Fiz. 60, 1969 (1971)</ref>
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| More recent work has established a relation of (super-)gravity theories near a spacelike singularity (BKL-limit)
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| with Lorentzian [[Kac-Moody algebra]]s, [[Weyl group]]s and hyperbolic
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| [[Coxeter group]]s.<ref>M. Henneaux, D. Persson, and P. Spindel, Living Reviews in Relativity 11, 1 (2008), 0710.1818)</ref><ref>M. Henneaux, D. Persson, and D. H. Wesley, Journal of High Energy Physics 2008, 052 (2008)</ref><ref>M. Henneaux, ArXiv e-prints (2008), 0806.4670</ref>
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| Other more recent work is concerned with the discrete nature of the
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| Kasner map and a continuous generalisation.<ref>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–+</ref><ref>N. J. Cornish and J. J. Levin, Phys. Rev. Lett. 78, 998 (1997)</ref><ref>N. J. Cornish and J. J. Levin, Phys. Rev. D 55, 7489 (1997)</ref>
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| ==Curvature of Bianchi spaces==
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| The Bianchi spaces have the property that their [[Ricci tensor]]s can be [[Separable differential equation|separated]] into a product of the [[basis vector]]s associated with the space and a coordinate-independent tensor.
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| For a given [[Metric (mathematics)|metric]]
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| :<math>ds^2 = \gamma_{ab} \xi^{(a)}_i \xi^{(b)}_k dx^i dx^k</math> (where <math>\xi^{(a)}_idx^i</math>
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| are [[differential form|1-forms]]), the Ricci curvature tensor <math>R_{ik}</math> is given by:
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| :<math>R_{ik} = R_{(a)(b)} \xi^{(a)}_i \xi^{(b)}_k</math>
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| :<math>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]</math>
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| where the indices on the structure constants are raised and lowered with <math>\gamma_{ab}</math> which is not a function of <math>x^i</math>.
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| ==See also==
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| *[[Table of Lie groups]]
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| *[[List of simple Lie groups]]
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| ==References== | |
| {{reflist}}
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| *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]
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| *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
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| *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
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| *Robert T. Jantzen, [http://www34.homepage.villanova.edu/robert.jantzen/bianchi/ Bianchi classification of 3-geometries: original papers in translation]
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| [[Category:Lie algebras]]
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| [[Category:Lie groups]]
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| [[Category:Physical cosmology]]
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