In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Bianchi classification provides a list of all real 3-dimensional
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s (
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
). 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 group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s serve as symmetry groups of 3-dimensional
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
s. It is named for
Luigi Bianchi
Luigi Bianchi (18 January 1856 – 6 June 1928) was an Italians, Italian mathematician. He was born in Parma, Emilia-Romagna, and died in Pisa. He was a leading member of the vigorous Italian school of algebraic geometry, geometric school which fl ...
, 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 algebra In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers.
Given a complex Lie algebra \mathfrak, its conjugate \overline is a complex Lie algebra with the same underlying real vector space but with i = \sqrt acting as -i inst ...
s.
Classification in dimension less than 3
* Dimension 0: The only Lie algebra is the
abelian Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
R
0.
* Dimension 1: The only Lie algebra is the abelian Lie algebra R
1, with
outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...
the multiplicative group of non-zero real numbers.
* Dimension 2: There are two Lie algebras:
** (1) The abelian Lie algebra R
2, with outer automorphism group
GL2(R).
** (2) The
solvable Lie algebra
In mathematics, a Lie algebra \mathfrak is solvable if its derived series terminates in the zero subalgebra. The ''derived Lie algebra'' of the Lie algebra \mathfrak is the subalgebra of \mathfrak, denoted
: mathfrak,\mathfrak/math>
that consist ...
of 2×2 upper triangular matrices of trace 0. It has trivial center and trivial outer automorphism group. The
associated Associated may refer to:
*Associated, former name of Avon, Contra Costa County, California
* Associated Hebrew Schools of Toronto, a school in Canada
*Associated Newspapers, former name of DMG Media, a British publishing company
See also
*Associati ...
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
is the
affine group
In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself.
It is a Lie group if is the real or complex field or quaternions.
Relat ...
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
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
* an ''inner'' semidirect product is a particular way in w ...
of R
2 and R, with R acting on R
2 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 R
3. The simply connected group has center R
3 and outer automorphism group GL
3(R). This is the case when ''M'' is 0.
*Type II: The
Heisenberg algebra
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
::\begin
1 & a & c\\
0 & 1 & b\\
0 & 0 & 1\\
\end
under the operation of matrix multiplication. Elements '' ...
, which is
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the class ...
and unimodular. The simply connected group has center R and outer automorphism group GL
2(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
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
.
*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 GL
2(R) of determinant +1 or −1. The matrix ''M'' has two equal eigenvalues, and is diagonalizable.
*Type VI: An infinite family: semidirect products of R
2 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 VI
0: This Lie algebra is the semidirect product of R
2 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 Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
, the group of isometries of 2-dimensional
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
. The simply connected group has trivial center and outer automorphism group the product of the
positive real numbers
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
with the
dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
of order 8.
*Type VII: An infinite family: semidirect products of R
2 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 VII
0: Semidirect product of R
2 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 ''sl''
2(R) of traceless 2 by 2 matrices, associated to the group
SL2(R). It is
simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
and unimodular. The simply connected group is not a matrix group; it is denoted by
, has center Z and its outer automorphism group has order 2.
*Type IX: The Lie algebra of the
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
''O''
3(R). It is denoted by
𝖘𝖔(3) and is simple and unimodular. The corresponding simply connected group is
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
; it has center of order 2 and trivial outer automorphism group, and is a
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a L ...
.
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
In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensi ...
. 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.
Structure constants
The three-dimensional Bianchi spaces each admit a set of three
Killing vector field
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal gene ...
s
which obey the following property:
:
where
, the "structure constants" of the group, form a
constant order-three tensor antisymmetric in its lower two indices. For any three-dimensional Bianchi space,
is given by the relationship
:
where
is the
Levi-Civita symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers , for some ...
,
is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
, and the vector
and
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
tensor
are described by the following table, where
gives the ''i''th
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of
; the parameter ''a'' runs over all positive
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s:
![Bianchi-classification](https://upload.wikimedia.org/wikipedia/commons/6/67/Bianchi-classification.svg)
The standard Bianchi classification can be derived from the structural constants in the following six steps:
#Due to the antisymmetry
, there are nine independent constants
. These can be equivalently represented by the nine components of an arbitrary constant matrix ''C''
''ab'':
where ε
''abd'' is the totally antisymmetric three-dimensional Levi-Civita symbol (ε
123 = 1). Substitution of this expression for
into the
Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the asso ...
, results in
#The structure constants can be transformed as:
Appearance of det A in this formula is due to the fact that the symbol ε
''abd'' transforms as tensor density:
, where έ
''mnd'' ≡ ε
''mnd''. By this transformation it is always possible to reduce the matrix ''C''
''ab'' to the form:
After such a choice, one still have the freedom of making triad transformations but with the restrictions
and
#Now, the Jacobi identities give only one constraint:
#If ''n''
1 ≠ 0 then ''C''
23 – ''C''
32 = 0 and by the remaining transformations with
, the 2 × 2 matrix
in ''C''
''ab'' can be made diagonal. Then
The diagonality condition for ''C''
''ab'' is preserved under the transformations with diagonal
. Under these transformations, the three parameters ''n''
1, ''n''
2, ''n''
3 change in the following way:
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 ''n''
1, ''n''
2, ''n''
3 (invariantly different in the sense that there is no way to pass from one to another by some transformation of the triad
), that is to the following different types of homogeneous spaces with diagonal matrix ''C''
''ab'':
#Consider now the case ''n''
1 = 0. It can also happen in that case that ''C''
23 – ''C''
32 = 0. This returns to the situation already analyzed in the previous step but with the additional condition ''n''
1 = 0. Now, all essentially different types for the sets ''n''
1, ''n''
2, ''n''
3 are (0, 1, 1), (0, 1, −1), (0, 0, 1) and (0, 0, 0). The first three repeat the types ''VII''
0, ''VI''
0, ''II''. Consequently, only one new type arises:
#The only case left is ''n''
1 = 0 and ''C''
23 – ''C''
32 ≠ 0. Now the 2 × 2 matrix
is non-symmetric and it cannot be made diagonal by transformations using
. However, its symmetric part can be diagonalized, that is the 3 × 3 matrix ''C''
''ab'' can be reduced to the form:
where ''a'' is an arbitrary number. After this is done, there still remains the possibility to perform transformations with diagonal
, under which the quantities ''n''
2, ''n''
3 and ''a'' change as follows:
These formulas show that for nonzero ''n''
2, ''n''
3, ''a'', the combination ''a''
2(''n''
2''n''
3)
−1 is an invariant quantity. By a choice of
, one can impose the condition ''a'' > 0 and after this is done, the choice of the sign of
permits one to change both signs of ''n''
2 and ''n''
3 simultaneously, that is the set (''n''
2 , ''n''
3) is equivalent to the set (−''n''
2,−''n''
3). It follows that there are the following four different possibilities:
For the first two, the number ''a'' can be transformed to unity by a choice of
the parameters
and
. 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:
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 ''VII''
0 is a particular case of ''VII'' corresponding to ''a'' = 0 while type ''VI''
0 is a particular case of ''VI'' corresponding also to ''a'' = 0.
Curvature of Bianchi spaces
The Bianchi spaces have the property that their
Ricci tensor
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
s can be
separated into a product of the
basis vector
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
s associated with the space and a coordinate-independent tensor.
For a given
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathema ...
:
:
(where
are
1-forms
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to eac ...
), the Ricci curvature tensor
is given by:
:
: