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 SL ( 2 , R ) ¯ {\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 The three-dimensional Bianchi spaces each admit a set of three Killing vector fields ξ i ( a ) {\displaystyle \xi _{i}^{(a)}} which obey the following property:
( ∂ ξ i ( c ) ∂ x k − ∂ ξ k ( c ) ∂ x i ) ξ ( a ) i ξ ( b ) k = C a b c {\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 C a b c {\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, C a b c {\displaystyle C_{\ ab}^{c}} is given by the relationship
C a b c = ε a b d n c d − δ a c a b + δ b c a a {\displaystyle C_{\ ab}^{c}=\varepsilon _{abd}n^{cd}-\delta _{a}^{c}a_{b}+\delta _{b}^{c}a_{a}}
where ε a b d {\displaystyle \varepsilon _{abd}} is the Levi-Civita symbol, δ a c {\displaystyle \delta _{a}^{c}} is the Kronecker delta, and the vector a a = ( a , 0 , 0 ) {\displaystyle a_{a}=(a,0,0)} and diagonal tensor n c d {\displaystyle n^{cd}} are described by the following table, where n ( i ) {\displaystyle n^{(i)}} gives the ith eigenvalue of n c d {\displaystyle n^{cd}} ; the parameter a runs over all positive real numbers:
The standard Bianchi classification can be derived from the structural constants in the following six steps:
Due to the antisymmetry C a b c = − C b a c {\displaystyle C_{ab}^{c}=-C_{ba}^{c}} , there are nine independent constants C a b c {\displaystyle C_{ab}^{c}} . These can be equivalently represented by the nine components of an arbitrary constant matrix Cab: C a b c = ε a b d C d c , {\displaystyle C_{ab}^{c}=\varepsilon _{abd}C^{dc},} where εabd is the totally antisymmetric three-dimensional Levi-Civita symbol (ε123 = 1). Substitution of this expression for C a b c {\displaystyle C_{ab}^{c}} into the Jacobi identity, results in ε a b d C b d C a c = 0. {\displaystyle \varepsilon _{abd}C^{bd}C^{ac}=0.}
The structure constants can be transformed as: C a b = ( det A ) − 1 A m a A n b C ´ m n . {\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 εabd transforms as tensor density: ε a b c = ( det A ) D a m D b n D c d ε ´ m n d {\displaystyle \varepsilon _{abc}=\left(\det {A}\right)D_{a}^{m}D_{b}^{n}D_{c}^{d}{\acute {\varepsilon }}_{mnd}} , where έmnd ≡ εmnd. By this transformation it is always possible to reduce the matrix Cab to the form: C a b = [ n 1 0 0 0 C 22 C 23 0 C 32 C 33 ] . {\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 A 2 1 = A 3 1 = 0 {\displaystyle A_{2}^{1}=A_{3}^{1}=0} and A 1 2 = A 1 3 = 0. {\displaystyle A_{1}^{2}=A_{1}^{3}=0.}
Now, the Jacobi identities give only one constraint: ( C 23 − C 32 ) n 1 = 0. {\displaystyle \left(C^{23}-C^{32}\right)n_{1}=0.}
If n1 ≠ 0 then C23 – C32 = 0 and by the remaining transformations with A b ¯ a ¯ ≠ 0 , a ¯ , b ¯ = 2 ¯ , 3 ¯ {\displaystyle A_{\bar {b}}^{\bar {a}}\neq 0,\quad {\bar {a}},{\bar {b}}={\bar {2}},{\bar {3}}} , the 2 × 2 matrix C a ¯ b ¯ {\displaystyle C^{{\bar {a}}{\bar {b}}}} in Cab can be made diagonal. Then C a b = [ n 1 0 0 0 n 2 0 0 0 n 3 ] . {\displaystyle C^{ab}={\begin{bmatrix}n_{1}&0&0\\0&n_{2}&0\\0&0&n_{3}\end{bmatrix}}.} The diagonality condition for Cab is preserved under the transformations with diagonal A b a {\displaystyle A_{b}^{a}} . Under these transformations, the three parameters n1, n2, n3 change in the following way: n a = ( A 1 1 A 2 2 A 3 3 ) ( A a a ) 2 n ´ a , no summation over a . {\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 na (if it is not zero) can be made equal to unity. Taking into account that the simultaneous change of sign of all na 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 e a a ¯ = A b ¯ a ¯ e a b ¯ {\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 Cab: B i a n c h i I X : ( n 1 , n 2 , n 3 ) = ( 1 , 1 , 1 ) , B i a n c h i V I I I : ( n 1 , n 2 , n 3 ) = ( 1 , 1 , − 1 ) , B i a n c h i V I I 0 : ( n 1 , n 2 , n 3 ) = ( 1 , 1 , 0 ) , B i a n c h i V I 0 : ( n 1 , n 2 , n 3 ) = ( 1 , − 1 , 0 ) , B i a n c h i I I : ( n 1 , n 2 , n 3 ) = ( 1 , 0 , 0 ) . {\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}}}
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: B i a n c h i I : ( n 1 , n 2 , n 3 ) = ( 0 , 0 , 0 ) . {\displaystyle Bianchi\ I\ :\ (n_{1},n_{2},n_{3})\ =\ (0,0,0).}
The only case left is n1 = 0 and C23 – C32 ≠ 0. Now the 2 × 2 matrix C a ¯ b ¯ ( a ¯ , b ¯ = 2 , 3 ) {\displaystyle C^{{\bar {a}}{\bar {b}}}({\bar {a}},{\bar {b}}=2,3)} is non-symmetric and it cannot be made diagonal by transformations using A b ¯ a ¯ ≠ 0 {\displaystyle A_{\bar {b}}^{\bar {a}}\neq 0} . However, its symmetric part can be diagonalized, that is the 3 × 3 matrix Cab can be reduced to the form: C a b = [ 0 0 0 0 n 2 a 0 − a n 3 ] , {\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 A b ¯ a ¯ {\displaystyle A_{\bar {b}}^{\bar {a}}} , under which the quantities n2, n3 and a change as follows: n 2 = ( A 1 1 A 2 2 A 3 3 ) − 1 ( A 2 2 ) 2 n ´ 2 , n 3 = ( A 1 1 A 2 2 A 3 3 ) − 1 ( A 3 3 ) 2 n ´ 3 , a = ( A 1 1 ) − 1 a ´ . {\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 A 1 1 {\displaystyle A_{1}^{1}} , one can impose the condition a > 0 and after this is done, the choice of the sign of A 3 3 ( A 2 2 ) − 1 {\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: ( a , n 2 , n 3 ) = ( a , 0 , 0 ) , ( a , 0 , 1 ) , ( a , 1 , 1 ) , ( a , 1 , − 1 ) . {\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 ofthe parameters A 1 1 {\displaystyle A_{1}^{1}} and A 3 3 ( A 2 2 ) − 1 {\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: B i a n c h i V : n 1 = 0 , ( a , n 2 , n 3 ) = ( 1 , 0 , 0 ) , B i a n c h i I V : n 1 = 0 , ( a , n 2 , n 3 ) = ( 1 , 0 , 1 ) , B i a n c h i V I I : n 1 = 0 , ( a , n 2 , n 3