Bianchi classification

In mathematics, the Bianchi classification provides a list of all real 3-dimensional Lie algebras (up to isomorphism). The classification contains 11 classes, 9 of which contain a single Lie algebra and two of which contain a continuum-sized family of Lie algebras. (Sometimes two of the groups are included in the infinite families, giving 9 instead of 11 classes.) The classification is important in geometry and physics, because the associated Lie groups serve as symmetry groups of 3-dimensional Riemannian manifolds. It is named for Luigi Bianchi, who worked it out in 1898.

The term "Bianchi classification" is also used for similar classifications in other dimensions and for classifications of complex Lie algebras.

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 multiplicative 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. It has trivial center and trivial outer automorphism group. The associated simply connected Lie group is the affine group of the line.

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: The Heisenberg algebra, which is nilpotent and unimodular. 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: 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.) It is solvable and not unimodular. 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: The algebra generated by [y,z] = 0, [x,y] = y, [x, z] = y + z. It is solvable and not unimodular. 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 diagonalizable.
• Type V: [y,z] = 0, [x,y] = y, [x, z] = z. Solvable and not unimodular. (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 diagonalizable.
• Type VI: An infinite family: semidirect products of R² by R, where the matrix M has non-zero distinct real eigenvalues with non-zero sum. The algebras are solvable and not unimodular. 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: 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 solvable and unimodular. It is the Lie algebra of the 2-dimensional Poincaré group, 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: An infinite family: semidirect products of R² by R, where the matrix M has non-real and non-imaginary eigenvalues. Solvable and not unimodular. The simply connected group has trivial center and outer automorphism group the non-zero reals.
• Type VII₀: Semidirect product of R² by R, where the matrix M has non-zero imaginary eigenvalues. Solvable and unimodular. 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: The Lie algebra sl₂(R) of traceless 2 by 2 matrices, associated to the group SL₂(R). It is simple and unimodular. The simply connected group is not a matrix group; it is denoted by ${\displaystyle {\overline {{\mbox{SL}}(2,\mathbf {R} )}}}$, has center Z and its outer automorphism group has order 2.
• Type IX: The Lie algebra of the orthogonal group O₃(R). It is denoted by 𝖘𝖔(3) and is simple and unimodular. The corresponding simply connected group is SU(2); it 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

Figure 1. The parameter space as a 3-plane (class A) and an orthogonal half 3-plane (class B) in R⁴ with coordinates (n⁽¹⁾, n⁽²⁾, n⁽³⁾, a), showing the canonical representatives of each Bianchi type.

The three-dimensional Bianchi spaces each admit a set of three Killing vector fields ${\displaystyle \xi _{i}^{(a)}}$ which obey the following property:

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

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

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

where ${\displaystyle \varepsilon _{abd}}$ is the Levi-Civita symbol, ${\displaystyle \delta _{a}^{c}}$ is the Kronecker delta, and the vector ${\displaystyle a_{a}=(a,0,0)}$ and diagonal tensor ${\displaystyle n^{cd}}$ are described by the following table, where ${\displaystyle n^{(i)}}$ gives the ith eigenvalue of ${\displaystyle n^{cd}}$;[1] the parameter a runs over all positive real numbers:

Bianchi type	${\displaystyle a}$	${\displaystyle n^{(1)}}$	${\displaystyle n^{(2)}}$	${\displaystyle n^{(3)}}$	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 ${\displaystyle a=1}$
IV	1	0	0	1
V	1	0	0	0	has a hyper-pseudosphere as a special case
VI0	0	1	-1	0
VIa	${\displaystyle a}$	0	1	-1	when ${\displaystyle a=1}$, equivalent to type III
VII0	0	1	1	0	has Euclidean space as a special case
VIIa	${\displaystyle a}$	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

The standard Bianchi classification can be derived from the structural constants in the following six steps:

1. Due to the antisymmetry ${\displaystyle C_{ab}^{c}=-C_{ba}^{c}}$, there are nine independent constants ${\displaystyle C_{ab}^{c}}$. These can be equivalently represented by the nine components of an arbitrary constant matrix C^ab:

   ${\displaystyle C_{ab}^{c}=\varepsilon _{abd}C^{dc},}$

   where ε_abd is the totally antisymmetric three-dimensional Levi-Civita symbol (ε₁₂₃ = 1). Substitution of this expression for ${\displaystyle C_{ab}^{c}}$ into the Jacobi identity, results in

   ${\displaystyle \varepsilon _{abd}C^{bd}C^{ac}=0.}$

2. The structure constants can be transformed as:

   ${\displaystyle C^{ab}=\left(\det {A}\right)^{-1}A_{m}^{a}A_{n}^{b}{\acute {C}}^{mn}.}$

   Appearance of det A in this formula is due to the fact that the symbol ε_abd transforms as tensor density: ${\displaystyle \varepsilon _{abc}=\left(\det {A}\right)D_{a}^{m}D_{b}^{n}D_{c}^{d}{\acute {\varepsilon }}_{mnd}}$, where έ_mnd ≡ ε_mnd. By this transformation it is always possible to reduce the matrix C^ab to the form:

   ${\displaystyle C^{ab}={\begin{bmatrix}n_{1}&0&0\\0&C^{22}&C^{23}\\0&C^{32}&C^{33}\end{bmatrix}}.}$

   After such a choice, one still have the freedom of making triad transformations but with the restrictions ${\displaystyle A_{2}^{1}=A_{3}^{1}=0}$ and ${\displaystyle A_{1}^{2}=A_{1}^{3}=0.}$
3. Now, the Jacobi identities give only one constraint:

   ${\displaystyle \left(C^{23}-C^{32}\right)n_{1}=0.}$

4. If n₁ ≠ 0 then C²³ – C³² = 0 and by the remaining transformations with ${\displaystyle A_{\bar {b}}^{\bar {a}}\neq 0,\quad {\bar {a}},{\bar {b}}={\bar {2}},{\bar {3}}}$, the 2 × 2 matrix ${\displaystyle C^{{\bar {a}}{\bar {b}}}}$ in C^ab can be made diagonal. Then

   ${\displaystyle C^{ab}={\begin{bmatrix}n_{1}&0&0\\0&n_{2}&0\\0&0&n_{3}\end{bmatrix}}.}$

   The diagonality condition for C^ab is preserved under the transformations with diagonal ${\displaystyle A_{b}^{a}}$. Under these transformations, the three parameters n₁, n₂, n₃ change in the following way:

   ${\displaystyle n_{a}=\left(A_{1}^{1}A_{2}^{2}A_{3}^{3}\right)\left(A_{a}^{a}\right)^{2}{\acute {n}}_{a},{\text{no summation over}}\ a.}$

   By these diagonal transformations, the modulus of any n_a (if it is not zero) can be made equal to unity. Taking into account that the simultaneous change of sign of all n_a produce nothing new, one arrives to the following invariantly different sets for the numbers n₁, n₂, n₃ (invariantly different in the sense that there is no way to pass from one to another by some transformation of the triad ${\displaystyle e_{a}^{\bar {a}}=A_{\bar {b}}^{\bar {a}}e_{a}^{\bar {b}}}$), that is to the following different types of homogeneous spaces with diagonal matrix C^ab:

   ${\displaystyle {\begin{matrix}Bianchi\ IX&:&(n_{1},n_{2},n_{3})&=&(1,1,1),\\Bianchi\ VIII&:&(n_{1},n_{2},n_{3})&=&(1,1,-1),\\Bianchi\ VII_{0}&:&(n_{1},n_{2},n_{3})&=&(1,1,0),\\Bianchi\ VI_{0}&:&(n_{1},n_{2},n_{3})&=&(1,-1,0),\\Bianchi\ II&:&(n_{1},n_{2},n_{3})&=&(1,0,0).\end{matrix}}}$

5. Consider now the case n₁ = 0. It can also happen in that case that C²³ – C³² = 0. This returns to the situation already analyzed in the previous step but with the additional condition n₁ = 0. Now, all essentially different types for the sets n₁, n₂, n₃ are (0, 1, 1), (0, 1, −1), (0, 0, 1) and (0, 0, 0). The first three repeat the types VII₀, VI₀, II. Consequently, only one new type arises:

   ${\displaystyle Bianchi\ I\$ :\ (n_{1},n_{2},n_{3})\ =\ (0,0,0).}

6. The only case left is n₁ = 0 and C²³ – C³² ≠ 0. Now the 2 × 2 matrix ${\displaystyle C^{{\bar {a}}{\bar {b}}}({\bar {a}},{\bar {b}}=2,3)}$ is non-symmetric and it cannot be made diagonal by transformations using ${\displaystyle A_{\bar {b}}^{\bar {a}}\neq 0}$. However, its symmetric part can be diagonalized, that is the 3 × 3 matrix C^ab can be reduced to the form:

   ${\displaystyle C^{ab}={\begin{bmatrix}0&0&0\\0&n_{2}&a\\0&-a&n_{3}\end{bmatrix}},}$

   where a is an arbitrary number. After this is done, there still remains the possibility to perform transformations with diagonal ${\displaystyle A_{\bar {b}}^{\bar {a}}}$, under which the quantities n₂, n₃ and a change as follows:

   ${\displaystyle n_{2}=\left(A_{1}^{1}A_{2}^{2}A_{3}^{3}\right)^{-1}\left(A_{2}^{2}\right)^{2}{\acute {n}}_{2},\quad n_{3}=\left(A_{1}^{1}A_{2}^{2}A_{3}^{3}\right)^{-1}\left(A_{3}^{3}\right)^{2}{\acute {n}}_{3},\quad a=\left(A_{1}^{1}\right)^{-1}{\acute {a}}.}$

   These formulas show that for nonzero n₂, n₃, a, the combination a²(n₂n₃)⁻¹ is an invariant quantity. By a choice of ${\displaystyle A_{1}^{1}}$, one can impose the condition a > 0 and after this is done, the choice of the sign of ${\displaystyle A_{3}^{3}\left(A_{2}^{2}\right)^{-1}}$ permits one to change both signs of n₂ and n₃ simultaneously, that is the set (n₂ , n₃) is equivalent to the set (−n₂,−n₃). It follows that there are the following four different possibilities:

   ${\displaystyle (a,n_{2},n_{3})=(a,0,0),(a,0,1),(a,1,1),(a,1,-1).}$

   For the first two, the number a can be transformed to unity by a choice of the parameters ${\displaystyle A_{1}^{1}}$ and ${\displaystyle A_{3}^{3}\left(A_{2}^{2}\right)^{-1}}$. For the second two possibilities, both of these parameters are already fixed and a remains an invariant and arbitrary positive number. Historically these four types of homogeneous spaces have been classified as:

   ${\displaystyle {\begin{matrix}Bianchi\ V&:&n_{1}=0,\ (a,n_{2},n_{3})&=&(1,0,0),\\Bianchi\ IV&:&n_{1}=0,\ (a,n_{2},n_{3})&=&(1,0,1),\\Bianchi\ VII&:&n_{1}=0,\ (a,n_{2},n_{3})&=&(a,1,1),\\Bianchi\ III&:&n_{1}=0,\ (a,n_{2},n_{3})&=&(1,1,-1),\\Bianchi\ VI&:&n_{1}=0,\ (a,n_{2},n_{3})&=&(a,1,-1).\end{matrix}}}$

   Type III is just a particular case of type VI corresponding to a = 1. Types VII and VI contain an infinity of invariantly different types of algebras corresponding to the arbitrariness of the continuous parameter a. Type VII₀ is a particular case of VII corresponding to a = 0 while type VI₀ is a particular case of VI corresponding also to a = 0.

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:

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

(where ${\displaystyle \xi _{i}^{(a)}dx^{i}}$　are 1-forms), the Ricci curvature tensor ${\displaystyle R_{ik}}$ is given by:

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

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

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

Cosmological application

In cosmology, this classification is used for a homogeneous spacetime of dimension 3+1. The 3-dimensional Lie group is as the symmetry group of the 3-dimensional spacelike slice, and the Lorentz metric satisfying the Einstein equation is generated by varying the metric components as a function of t. The Friedmann–Lemaître–Robertson–Walker metrics are isotropic, which are particular cases of types I, V, ${\displaystyle \scriptstyle {\text{VII}}_{h}}$ 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; 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]

See also

• Table of Lie groups
• List of simple Lie groups

References

1. Lev Landau & Evgeny Lifshitz (1980), Course of Theoretical Physics vol. 2: The Classical Theory of Fields, Butterworth-Heinemann, ISBN 978-0-7506-2768-9
2. Robert Wald, General Relativity, University of Chicago Press (1984). ISBN 0-226-87033-2, (chapt 7.2, pages 168–179)
3. V. A. Belinskii, I. M. Khalatnikov, and E. M. Lifshitz, Zh. Eksp. Teor. Fiz. 62, 1606 (1972)
4. V. A. Belinskii, I. M. Khalatnikov, and E. M. Lifshitz, Zh. Eksp. Teor. Fiz. 60, 1969 (1971)
5. Henneaux, Marc; Persson, Daniel; Spindel, Philippe (2008-04-24). "Spacelike Singularities and Hidden Symmetries of Gravity". Living Reviews in Relativity. 11 (1): 1. arXiv:0710.1818. Bibcode:2008LRR....11....1H. doi:10.12942/lrr-2008-1. ISSN 2367-3613. PMC 5255974. PMID 28179821.
6. Henneaux, Marc; Persson, Daniel; Wesley, Daniel H (2008-09-10). "Coxeter group structure of cosmological billiards on compact spatial manifolds". Journal of High Energy Physics. 2008 (9): 052. arXiv:0805.3793. Bibcode:2008JHEP...09..052H. doi:10.1088/1126-6708/2008/09/052. ISSN 1029-8479.
7. M. Henneaux, ArXiv e-prints (2008), 0806.4670
8. 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–+
9. Cornish, Neil J.; Levin, Janna J. (1997-02-10). "The Mixmaster Universe is Chaotic". Physical Review Letters. 78 (6): 998–1001. arXiv:gr-qc/9605029. Bibcode:1997PhRvL..78..998C. doi:10.1103/physrevlett.78.998. ISSN 0031-9007.
10. Cornish, Neil J.; Levin, Janna J. (1997-06-15). "Mixmaster universe: A chaotic Farey tale". Physical Review D. American Physical Society (APS). 55 (12): 7489–7510. arXiv:gr-qc/9612066. Bibcode:1997PhRvD..55.7489C. doi:10.1103/physrevd.55.7489. ISSN 0556-2821.

• 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