Editing Lorentz transformation (section)

====The Lie algebra so(3,1)====

Lorentz generators can be added together, or multiplied by real numbers, to obtain more Lorentz generators. In other words, the [[set (mathematics)|set]] of all Lorentz generators
<math display="block">V = \{ \boldsymbol{\zeta} \cdot\mathbf{K} + \boldsymbol{\theta} \cdot\mathbf{J}  \} </math>
together with the operations of ordinary [[matrix addition]] and [[matrix multiplication#Scalar multiplication|multiplication of a matrix by a number]], forms a [[vector space]] over the real numbers.<ref group=nb>Until now the term "vector" has exclusively referred to "[[Euclidean vector]]", examples are position {{math|'''r'''}}, velocity {{math|'''v'''}}, etc. The term "vector" applies much more broadly than Euclidean vectors, row or column vectors, etc., see [[linear algebra]] and [[vector space]] for details. The generators of a Lie group also form a vector space over a [[field (mathematics)|field]] of numbers (e.g. [[real number]]s, [[complex number]]s), since a [[linear combination]] of the generators is also a generator. They just live in a different space to the position vectors in ordinary 3-dimensional space.</ref> The generators {{math|''J''{{sub|''x''}}, ''J''{{sub|''y''}}, ''J''{{sub|''z''}}, ''K''{{sub|''x''}}, ''K''{{sub|''y''}}, ''K''{{sub|''z''}}}} form a [[basis (linear algebra)|basis]] set of ''V'', and the components of the axis-angle and rapidity vectors, {{math|''θ''{{sub|''x''}}, ''θ''{{sub|''y''}}, ''θ''{{sub|''z''}}, ''ζ''{{sub|''x''}}, ''ζ''{{sub|''y''}}, ''ζ''{{sub|''z''}}}}, are the [[coordinate vector|coordinate]]s of a Lorentz generator with respect to this basis.<ref group=nb>In ordinary 3-dimensional [[position space]], the position vector {{math|'''r''' {{=}} ''x'''''e'''{{sub|''x''}} + ''y'''''e'''{{sub|''y''}} + ''z'''''e'''{{sub|''z''}}}} is expressed as a linear combination of the Cartesian unit vectors {{math|'''e'''{{sub|''x''}}, '''e'''{{sub|''y''}}, '''e'''{{sub|''z''}}}} which form a basis, and the Cartesian coordinates {{math|''x, y, z''}} are coordinates with respect to this basis.</ref>

Three of the [[commutation relation]]s of the Lorentz generators are
<math display="block">[ J_x, J_y ] =  J_z \,,\quad [ K_x, K_y ] = -J_z \,,\quad [ J_x, K_y ] =  K_z \,, </math>
where the bracket {{math|1=[''A'', ''B''] = ''AB'' − ''BA''}} is known as the ''[[commutator]]'', and the other relations can be found by taking [[cyclic permutation]]s of {{mvar|x}}, {{mvar|y}}, {{mvar|z}} components (i.e. change {{mvar|x}} to {{mvar|y}}, {{mvar|y}} to {{mvar|z}}, and {{mvar|z}} to {{mvar|x}}, repeat).

These commutation relations, and the vector space of generators, fulfill the definition of the [[Lie algebra]] <math>\mathfrak{so}(3, 1)</math>. In summary, a Lie algebra is defined as a [[vector space]] ''V'' over a [[field (mathematics)|field]] of numbers, and with a [[binary operation]] [ , ] (called a [[Lie bracket]] in this context) on the elements of the vector space, satisfying the axioms of [[Bilinear map|bilinearity]], [[alternatization]], and the [[Jacobi identity]]. Here the operation [ , ] is the commutator which satisfies all of these axioms, the vector space is the set of Lorentz generators ''V'' as given previously, and the field is the set of real numbers.

Linking terminology used in mathematics and physics: A group generator is any element of the Lie algebra. A group parameter is a component of a coordinate vector representing an arbitrary element of the Lie algebra with respect to some basis. A basis, then, is a set of generators being a basis of the Lie algebra in the usual vector space sense.

The [[exponential map (Lie theory)|exponential map]] from the Lie algebra to the Lie group,
<math display="block">\exp \, : \, \mathfrak{so}(3,1) \to \mathrm{SO}(3,1),</math>
provides a one-to-one correspondence between small enough neighborhoods of the origin of the Lie algebra and neighborhoods of the identity element of the Lie group. In the case of the Lorentz group, the exponential map is just the [[matrix exponential]]. Globally, the exponential map is not one-to-one, but in the case of the Lorentz group, it is [[surjective function|surjective]] (onto). Hence any group element in the connected component of the identity can be expressed as an exponential of an element of the Lie algebra.