Null vector

<h2 id="split-algebras">Split algebras</h2>
A composition algebra with a null vector is a split algebra.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
In a <a href="/facts/Composition_algebra/Y230strm">composition algebra</a> (A, +, ×, *), the quadratic form is q(x) = x x*. When x is a null vector then there is no multiplicative inverse for x, and since x ≠ 0, A is not a <a href="/facts/Division_algebra/xp5Bepx3">division algebra</a>.
In the <a href="/facts/Cayley%E2%80%93Dickson_construction/aDmlkB9m">Cayley–Dickson construction</a>, the split algebras arise in the series <a href="/facts/Bicomplex_number/BqFqexwX">bicomplex numbers</a>, <a href="/facts/Biquaternion/YsI2czSu">biquaternions</a>, and <a href="/facts/Bioctonion/cPjH6jY1">bioctonions</a>, which uses the <a href="/facts/Complex_number/2w7mqI7e">complex number</a> field 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbb {C} }
 
 as the foundation of this doubling construction due to <a href="/facts/L._E._Dickson/zJ3rPHDJ">L. E. Dickson</a> (1919). In particular, these algebras have two <a href="/facts/Imaginary_unit/1SCllQlU">imaginary units</a>, which commute so their product, when squared, yields +1:

(
        h
        i
        
          )
          
            2
          
        
        =
        
          h
          
            2
          
        
        
          i
          
            2
          
        
        =
        (
        −
        1
        )
        (
        −
        1
        )
        =
        +
        1.
      
    
    {\displaystyle (hi)^{2}=h^{2}i^{2}=(-1)(-1)=+1.}
  
 Then

(
 1
 +
 h
 i
 )
 (
 1
 +
 h
 i
 
 )
 
 ∗
 
 
 =
 (
 1
 +
 h
 i
 )
 (
 1
 −
 h
 i
 )
 =
 1
 −
 (
 h
 i
 
 )
 
 2
 
 
 =
 0
 
 
 {\displaystyle (1+hi)(1+hi)^{*}=(1+hi)(1-hi)=1-(hi)^{2}=0}
 
 so 1 + hi is a null vector.
The real subalgebras, <a href="/facts/Split_complex_number/DtavqChA">split complex numbers</a>, <a href="/facts/Split_quaternion/1nltLGbk">split quaternions</a>, and <a href="/facts/Split-octonion/gl1z7Y9R">split-octonions</a>, with their null cones representing the light tracking into and out of 0 ∈ A, suggest <a href="/facts/Spacetime_topology/GFvEtjt6">spacetime topology</a>.

<h2 id="examples">Examples</h2>
The <a href="/facts/Minkowski_space/vDkfkQJn">light-like</a> vectors of <a href="/facts/Minkowski_space/vDkfkQJn">Minkowski space</a> are null vectors.
The four <a href="/facts/Linearly_independent/8JrHfIIa">linearly independent</a> <a href="/facts/Biquaternion/YsI2czSu">biquaternions</a> l = 1 + hi, n = 1 + hj, m = 1 + hk, and m∗ = 1 – hk are null vectors and { l, n, m, m∗ } can serve as a <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a> for the subspace used to represent <a href="/facts/Spacetime/uRIN1x4b">spacetime</a>. Null vectors are also used in the <a href="/facts/Newman%E2%80%93Penrose_formalism/U18aSKry">Newman–Penrose formalism</a> approach to spacetime manifolds.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
In the <a href="/facts/Verma_module/Er0U0eEE">Verma module</a> of a <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> there are null vectors.

<ul><li>Dubrovin, B. A.; <a href="/facts/Anatoly_Fomenko/mevUTjJY">Fomenko, A. T.</a>; <a href="/facts/Sergei_Novikov_(mathematician)/hqMGDRul">Novikov, S. P.</a> (1984). <a href="https://archive.org/details/moderngeometryme000dubr">Modern Geometry: Methods and Applications</a>. Translated by Burns, Robert G. Springer. p. <a href="https://archive.org/details/moderngeometryme000dubr/page/50">50</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-90872-2.</li>
<li>Shaw, Ronald (1982). <a href="https://books.google.com/books?id=C6DgAAAAMAAJ">Linear Algebra and Group Representations</a>. Vol. 1. <a href="/facts/Academic_Press/qxoKjzFy">Academic Press</a>. p. 151. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-639201-3.</li>
<li><a href="/facts/Eric_Harold_Neville/CwK2ex84">Neville, E. H. (Eric Harold)</a> (1922). <a href="https://archive.org/details/prolegomenatoana00nevi">Prolegomena to Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions</a>. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. p. <a href="https://archive.org/details/prolegomenatoana00nevi/page/204">204</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Emil Artin (1957) Geometric Algebra, isotropic <a href="/wiki/Emil_Artin" target="_blank">/wiki/Emil_Artin</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Arthur A. Sagle & Ralph E. Walde (1973) Introduction to Lie Groups and Lie Algebras, page 197, Academic Press <a href="/wiki/Academic_Press" target="_blank">/wiki/Academic_Press</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Patrick Dolan (1968) A Singularity-free solution of the Maxwell-Einstein Equations, Communications in Mathematical Physics 9(2):161–8, especially 166, link from Project Euclid <a href="http://projecteuclid.org/euclid.cmp/1103840725" target="_blank">http://projecteuclid.org/euclid.cmp/1103840725</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Null vector open-in-new

Null vector