Bianchi classification

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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.

Contents

[edit] 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.

[edit] 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.

[edit] 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 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 aa = (a,0,0) and diagonal tensor ncd are described by the following table, where n(i) gives the ith eigenvalue of ncd[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 VIa 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
VIa a 0 1 -1 when a = 1, equivalent to type III
VII0 0 1 1 0
VIIa a 0 1 1
VIII 0 1 1 -1
IX 0 1 1 1 has a hypersphere as a special case

[edit] 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.[2]

[edit] Curvature of Bianchi spaces

The Bianchi spaces have the property that their Ricci tensors can can be separated into a product of the Killing 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 Rik 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 γab which is not a function of xi.

[edit] See also

[edit] References

  1. ^ Lev Landau and Evgeny Lifshitz, Course of Theoretical Physics vol. 2: The Classical Theory of Fields, Butterworth-Heinemann, 1980.
  2. ^ Robert Wald, General Relativity, University of Chicago Press (1984). ISBN 0226870332, (chapt 7.2, pages 168 - 179)
  • 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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