Bianchi classification

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]


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Classification De Bianchi — La classification de Bianchi est une classification des algèbres de Lie réelles de dimension 3, donnée par Luigi Bianchi. Classification de Bianchi Type Description Exemple Groupe de Lie Matrice Type I Abélienne R³, muni d un crochet nul R³ comme …   Wikipédia en Français

  • Classification de bianchi — La classification de Bianchi est une classification des algèbres de Lie réelles de dimension 3, donnée par Luigi Bianchi. Classification de Bianchi Type Description Exemple Groupe de Lie Matrice Type I Abélienne R³, muni d un crochet nul R³ comme …   Wikipédia en Français

  • Classification de Bianchi — La classification de Bianchi est une classification des algèbres de Lie réelles de dimension 3, donnée par Luigi Bianchi. Classification de Bianchi Type Description Exemple Groupe de Lie Matrice Type I Abélienne R³, muni d un crochet nul R³ comme …   Wikipédia en Français

  • Bianchi cycling team — Infobox Cycling team teamname=Bianchi code=TBI base= ITA founded=1899 disbanded= manager=Jacques Hanegraaf (2003) techdirector= ds1 =Giovanni Tragella (1953 1954) ds2 =Franco Aguggini] (1956 1959) ds3 =Vittorio Adorni (1973) ds4 =Giancarlo… …   Wikipedia

  • Luigi Bianchi — Pour les articles homonymes, voir Bianchi. Luigi Bianchi, né le 18 janvier 1859 à Parme en Italie, mort le 6 juin 1928 à Pise, fut un des mathématiciens italiens meneurs de l’école de géométrie italienne de la fin du XIXe siècle et du début… …   Wikipédia en Français

  • Luigi Bianchi — (January 18 1856 June 6 1928) was an Italian mathematician. He was born in Parma, Emilia Romagna, and died in Pisa. He was a leading member of the vigorous geometric school which flourished in Italy during the later years of the 19th century and… …   Wikipedia

  • Young rider classification in the Giro d'Italia — White Jersey Award details Sport Road bicycle racing Competition Giro d Italia Given for Best young rider Local name(s) Maglia bianca …   Wikipedia

  • Landslide classification — There have been known various classifications of landslides and other types of mass wasting.For example, the McGraw Hill Encyclopedia of Science and Technology distinguishes the following types of landslies: *fall (by undercutting) *fall (by… …   Wikipedia

  • Nicolò Cesa-Bianchi — (Italian pronunciation: [nikɔˈlɔ]; born 18.6.1963. Milan, Italy) is a computer scientist and Professor of Computer Science at the Dipartimento di Scienze dell Informazione of the University of Milan.[1] He is a researcher in the field of… …   Wikipedia

  • Contributors to general relativity — General relativity Introduction Mathematical formulation Resources Fundamental concepts …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”