page_title: Lorentz group
section_title: Subgroups of the Lorentz group
hierarchical_section_title: Lorentz group / Subgroups of the Lorentz group
context_page_description: In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.
  For example, the following laws, equations, and theories respect Lorentz symmetry:
  The kinematical laws of special relativity
  Maxwell's field equations in the theory of electromagnetism
  The Dirac equation in the theory of the electron
  The Standard model of particle physics
  The Lorentz group expresses the fundamental symmetry of space and time of all known fundamental laws of nature. In general relativity physics, in cases involving small enough regions of spacetime where gravitational variances are negligible, physical laws are Lorentz invariant in the same manner as that of special relativity physics.
context_section_description: The subalgebras of the Lie algebra of the Lorentz group can be enumerated, up to conjugacy, from which we can list the closed subgroups of the restricted Lorentz group, up to conjugacy.  (See the book by Hall cited below for the details.)  We can readily express the result in terms of the generating set given in the table above.
  The one-dimensional subalgebras of course correspond to the four conjugacy classes of elements of the Lorentz group:
   generates a one-parameter subalgebra of parabolics SO(0,1),
   generates a one-parameter subalgebra of boosts SO(1,1),
   generates a one-parameter of rotations SO(2),
   (for any ) generates a one-parameter subalgebra of loxodromic transformations.
  (Strictly speaking the last corresponds to infinitely many classes, since distinct  give different classes.)
  The two-dimensional subalgebras are:
   generate an abelian subalgebra consisting entirely of parabolics,
   generate a nonabelian subalgebra isomorphic to the Lie algebra of the affine group A(1),
   generate an abelian subalgebra consisting of boosts, rotations, and loxodromics all sharing the same pair of fixed points.
  The three-dimensional subalgebras are:
   generate a Bianchi V subalgebra, isomorphic to the Lie algebra of Hom(2), the group of euclidean homotheties,
   generate a Bianchi VII_0 subalgebra, isomorphic to the Lie algebra of E(2), the euclidean group,
  , where , generate a Bianchi VII_a subalgebra,
   generate a Bianchi VIII subalgebra, isomorphic to the Lie algebra of SL(2,R), the group of isometries of the hyperbolic plane,
   generate a Bianchi IX subalgebra, isomorphic to the Lie algebra of SO(3), the rotation group.
  (Here, the Bianchi types refer to the classification of three-dimensional Lie algebras by the Italian mathematician Luigi Bianchi.)
  The four-dimensional subalgebras are all conjugate to
   generate a subalgebra isomorphic to the Lie algebra of Sim(2), the group of Euclidean similitudes.
  The subalgebras form a lattice (see the figure), and each subalgebra generates by exponentiation a closed subgroup of the restricted Lie group.  From these, all subgroups of the Lorentz group can be constructed, up to conjugation, by multiplying by one of the elements of the Klein four-group.
  As with any connected Lie group, the coset spaces of the closed subgroups of the restricted Lorentz group, or homogeneous spaces, have considerable mathematical interest. A few, brief descriptions:
  The group Sim(2) is the stabilizer of a null line, i.e., of a point on the Riemann sphere—so the homogeneous space SO⁺(1,3)/Sim(2) is the Kleinian geometry that represents conformal geometry on the sphere S.²
  The (identity component of the) Euclidean group SE(2) is the stabilizer of a null vector, so the homogeneous space SO⁺(1,3)/SE(2) is the momentum space of a massless particle; geometrically, this Kleinian geometry represents the degenerate geometry of the light cone in Minkowski spacetime.
  The rotation group SO(3) is the stabilizer of a timelike vector, so the homogeneous space SO⁺(1,3)/SO(3) is the momentum space of a massive particle; geometrically, this space is none other than three-dimensional hyperbolic space H³.
caption: The lattice of subalgebras of the Lie algebra SO(1,3), up to conjugacy.
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