Lorentz Group - Subgroups of The Lorentz Group

Subgroups of The Lorentz Group

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. We briefly describe some of them here:

• 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 which represents conformal geometry on the sphere S2,
• 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.