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Null vector

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(Redirected from Split algebra) Vector on which a quadratic form is zero This article is about zeros of a quadratic form. For the zero element in a vector space, see Zero vector. For null vectors in Minkowski space, see Minkowski space § Causal structure.
A null cone where q ( x , y , z ) = x 2 + y 2 − z 2 . {\displaystyle q(x,y,z)=x^{2}+y^{2}-z^{2}.}

In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x of X for which q(x) = 0.

In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector.

A quadratic space (X, q) which has a null vector is called a pseudo-Euclidean space. The term isotropic vector v when q(v) = 0 has been used in quadratic spaces, and anisotropic space for a quadratic space without null vectors.

A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces A and B, X = A + B, where q is positive-definite on A and negative-definite on B. The null cone, or isotropic cone, of X consists of the union of balanced spheres: ⋃ r ≥ 0 { x = a + b : q ( a ) = − q ( b ) = r ,     a , b ∈ B } . {\displaystyle \bigcup _{r\geq 0}\{x=a+b:q(a)=-q(b)=r,\ \ a,b\in B\}.} The null cone is also the union of the isotropic lines through the origin.

Split algebras

A composition algebra with a null vector is a split algebra.

In a composition algebra (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 division algebra.

In the Cayley–Dickson construction, the split algebras arise in the series bicomplex numbers, biquaternions, and bioctonions, which uses the complex number field C {\displaystyle \mathbb {C} } as the foundation of this doubling construction due to L. E. Dickson (1919). In particular, these algebras have two imaginary units, 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, split complex numbers, split quaternions, and split-octonions, with their null cones representing the light tracking into and out of 0 ∈ A, suggest spacetime topology.

Examples

The light-like vectors of Minkowski space are null vectors.

The four linearly independent biquaternions 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 basis for the subspace used to represent spacetime. Null vectors are also used in the Newman–Penrose formalism approach to spacetime manifolds.

In the Verma module of a Lie algebra there are null vectors.

References

  1. Emil Artin (1957) Geometric Algebra, isotropic
  2. Arthur A. Sagle & Ralph E. Walde (1973) Introduction to Lie Groups and Lie Algebras, page 197, Academic Press
  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
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