| width=15 | × !width=15 | 1 !width=15 | i !width=15 | j !width=15 | k |
|---|---|---|---|---|---|
| 1 | 1 | i | j | k | |
| i | i | −1 | k | −j | |
| j | j | −k | 1 | −i | |
| k | k | j | i | 1 |
In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers.
After introduction in the 20th century of coordinate-free definitions of rings and algebras, it was proved that the algebra of split-quaternions is isomorphic to the ring of the real matrices. So the study of split-quaternions can be reduced to the study of real matrices, and this may explain why there are few mentions of split-quaternions in the mathematical literature of the 20th and 21st centuries.
The split-quaternions are the linear combinations (with real coefficients) of four basis elements that satisfy the following product rules:
,
,
,
.By associativity, these relations imply
,
,and also .
So, the split-quaternions form a real vector space of dimension four with as a basis. They form also a noncommutative ring, by extending the above product rules by distributivity to all split-quaternions.
Let consider the square matrices
\begin{align} \boldsymbol{1}=\begin{pmatrix}1&0\\0&1\end{pmatrix}, &\boldsymbol{i}=\begin{pmatrix}0&1\\-1&0\end{pmatrix},\\ \boldsymbol{j}=\begin{pmatrix}0&1\\1&0\end{pmatrix}, &\boldsymbol{k}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}. \end{align}
\boldsymbol{1},\boldsymbol{i},\boldsymbol{j},\boldsymbol{k}
The above multiplication rules imply that the eight elements form a group under this multiplication, which is isomorphic to the dihedral group D4, the symmetry group of a square. In fact, if one considers a square whose vertices are the points whose coordinates are or, the matrix
\boldsymbol{i}
\boldsymbol{j}
\boldsymbol{k}
Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real associative algebra. But like the real algebra of 2×2 matrices – and unlike the real algebra of quaternions – the split-quaternions contain nontrivial zero divisors, nilpotent elements, and idempotents. (For example, is an idempotent zero-divisor, and is nilpotent.) As an algebra over the real numbers, the algebra of split-quaternions is isomorphic to the algebra of 2×2 real matrices by the above defined isomorphism.
This isomorphism allows identifying each split-quaternion with a 2×2 matrix. So every property of split-quaternions corresponds to a similar property of matrices, which is often named differently.
The conjugate of a split-quaternion, is . In term of matrices, the conjugate is the cofactor matrix obtained by exchanging the diagonal entries and changing the sign of the other two entries.
The product of a split-quaternion with its conjugate is the isotropic quadratic form:
N(q)=qq*=w2+x2-y2-z2,
The real part of a split-quaternion is . It equals the trace of associated matrix.
The norm of a product of two split-quaternions is the product of their norms. Equivalently, the determinant of a product of matrices is the product of their determinants. This property means that split-quaternions form a composition algebra. As there are nonzero split-quaternions having a zero norm, split-quaternions form a "split composition algebra" – hence their name.
A split-quaternion with a nonzero norm has a multiplicative inverse, namely . In terms of matrices, this is equivalent to the Cramer rule that asserts that a matrix is invertible if and only its determinant is nonzero, and, in this case, the inverse of the matrix is the quotient of the cofactor matrix by the determinant.
The isomorphism between split-quaternions and 2×2 real matrices shows that the multiplicative group of split-quaternions with a nonzero norm is isomorphic with
\operatorname{GL}(2,R),
\operatorname{SL}(2,R).
Geometrically, the split-quaternions can be compared to Hamilton's quaternions as pencils of planes. In both cases the real numbers form the axis of a pencil. In Hamilton quaternions there is a sphere of imaginary units, and any pair of antipodal imaginary units generates a complex plane with the real line. For split-quaternions there are hyperboloids of hyperbolic and imaginary units that generate split-complex or ordinary complex planes, as described below in § Stratification.
There is a representation of the split-quaternions as a unital associative subalgebra of the matrices with complex entries. This representation can be defined by the algebra homomorphism that maps a split-quaternion to the matrix
\begin{pmatrix}w+xi&y+zi\\y-zi&w-xi\end{pmatrix}.
The image of this homomorphism is the matrix ring formed by the matrices of the form
\begin{pmatrix}u&v\ v*&u*\end{pmatrix},
*
This homomorphism maps respectively the split-quaternions on the matrices
\begin{pmatrix}i&0\\0&-i\end{pmatrix}, \begin{pmatrix}0&1\\1&0\end{pmatrix}, \begin{pmatrix}0&i\\-i&0\end{pmatrix}.
The proof that this representation is an algebra homomorphism is straightforward but requires some boring computations, which can be avoided by starting from the expression of split-quaternions as real matrices, and using matrix similarity. Let be the matrix
S=\begin{pmatrix}1&i\\i&1\end{pmatrix}.
M\mapstoS-1MS.
It follows almost immediately that for a split quaternion represented as a complex matrix, the conjugate is the matrix of the cofactors, and the norm is the determinant.
With the representation of split quaternions as complex matrices. the matrices of quaternions of norm are exactly the elements of the special unitary group SU(1,1). This is used for in hyperbolic geometry for describing hyperbolic motions of the Poincaré disk model.
Split-quaternions may be generated by modified Cayley–Dickson construction[1] similar to the method of L. E. Dickson and Adrian Albert. for the division algebras C, H, and O. The multiplication ruleis used when producing the doubled product in the real-split cases. The doubled conjugate
(a,b)*=(a*,-b),
q=(a,b)=((w+zj),(y+xj)),
then
In this section, the real subalgebras generated by a single split-quaternion are studied and classified.
Let be a split-quaternion. Its real part is . Let be its nonreal part. One has, and therefore
p2=w2+2wq-N(q).
The structure of the subalgebra
R[p]
R[p]=R[q]=\{a+bq\mida,b\inR\},
R
The nonreal elements of
R[p]
a\inR.
Three cases have to be considered, which are detailed in the next subsections.
With above notation, if
q2=0,
x2-y2-z2=0.
R
p=w+ai+a\cos(t)j+a\sin(t)k.
This is a parametrization of all split-quaternions whose nonreal part is nilpotent.
This is also a parameterization of these subalgebras by the points of a circle: the split-quaternions of the form
i+\cos(t)j+\sin(t)k
The algebra generated by a nilpotent element is isomorphic to
R[X]/\langleX2\rangle
This is the case where . Letting one has
q2=-q*q=N(q)=n2=x2-y2-z2.
x2-y2-z2=1.
p=w+n\cosh(u)i+n\sinh(u)\cos(t)j+n\sinh(u)\sin(t)k.
This is a parametrization of all split-quaternions whose nonreal part has a positive norm.
This is also a parameterization of the corresponding subalgebras by the pairs of opposite points of a hyperboloid of two sheets: the split-quaternions of the form
\cosh(u)i+\sinh(u)\cos(t)j+\sinh(u)\sin(t)k
The algebra generated by a split-quaternion with a nonreal part of positive norm is isomorphic to
R[X]/\langleX2+1\rangle
\Complex
This is the case where . Letting one has
q2=-q*q=N(q)=-n2=x2-y2-z2.
p=w+n\sinh(u)i+n\cosh(u)\cos(t)j+n\cosh(u)\sin(t)k.
This is a parametrization of all split-quaternions whose nonreal part has a negative norm.
This is also a parameterization of the corresponding subalgebras by the pairs of opposite points of a hyperboloid of one sheet: the split-quaternions of the form
\sinh(u)i+\cosh(u)\cos(t)j+\cosh(u)\sin(t)k
The algebra generated by a split-quaternion with a nonreal part of negative norm is isomorphic to
R[X]/\langleX2-1\rangle
R2
As seen above, the purely nonreal split-quaternions of norm and form respectively a hyperboloid of one sheet, a hyperboloid of two sheets and a circular cone in the space of non real quaternions.
These surfaces are pairwise asymptote and do not intersect. Their complement consist of six connected regions:
N(q)>1
0<N(q)<1
-1<N(q)<0
N(q)<-1
This stratification can be refined by considering split-quaternions of a fixed norm: for every real number the purely nonreal split-quaternions of norm form an hyperboloid. All these hyperboloids are asymptote to the above cone, and none of these surfaces intersect any other. As the set of the purely nonreal split-quaternions is the disjoint union of these surfaces, this provides the desired stratification.
Split quaternions have been applied to colour balance[2] The model refers to the Jordan algebra of symmetric matrices representing the algebra. The model reconciles trichromacy with Hering's opponency and uses the Cayley–Klein model of hyperbolic geometry for chromatic distances.
The coquaternions were initially introduced (under that name)[3] in 1849 by James Cockle in the London - Edinburgh - Dublin Philosophical Magazine. The introductory papers by Cockle were recalled in the 1904 Bibliography[4] of the Quaternion Society.
Alexander Macfarlane called the structure of split-quaternion vectors an exspherical system when he was speaking at the International Congress of Mathematicians in Paris in 1900.[5] Macfarlane considered the "hyperboloidal counterpart to spherical analysis" in a 1910 article "Unification and Development of the Principles of the Algebra of Space" in the Bulletin of the Quaternion Society.[6]
The unit sphere was considered in 1910 by Hans Beck.[7] For example, the dihedral group appears on page 419. The split-quaternion structure has also been mentioned briefly in the Annals of Mathematics.[8] [9]