Split-quaternion

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 also equivalent to the Clifford algebras over the real numbers, and so the study of split-quaternions is also subsumed in the study of Clifford algebra and geometric algebra in the mathematics and physics literature.

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 also equivalent to the Clifford algebras over the real numbers, and so the study of split-quaternions is also subsumed in the study of Clifford algebra and geometric algebra in the mathematics and physics literature. \boldsymbol{1} =\begin{pmatrix}1&0\\0&1\end{pmatrix},\qquad&\boldsymbol{i} =\begin{pmatrix}0&1\\-1&0\end{pmatrix},\\ \boldsymbol{j} =\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad&\boldsymbol{k} =\begin{pmatrix}1&0\\0&-1\end{pmatrix}. \end{align} satisfy the same multiplication table as the corresponding split-quaternions. As these matrices form a basis of the two-by-two matrices, the unique linear function that maps to \boldsymbol{1}, \boldsymbol{i}, \boldsymbol{j}, \boldsymbol{k} (respectively) induces an algebra isomorphism from the split-quaternions to the two-by-two real matrices. The above multiplication rules imply that the eight elements form a group under this multiplication, which is isomorphic to the symmetry group of a square D 4 , the dihedral group of order 8. To see this, consider a square whose vertices are the points whose x\text{,} \, y coordinates are or so that the vertices \begin{pmatrix} x\\ y \end{pmatrix} of the square listed CW from azimuth +45° are \bigg \{ \mathbf x_1=\begin{pmatrix} +1\\ +1 \end{pmatrix} \text{,} \; \mathbf x_2=\begin{pmatrix} +1\\ -1 \end{pmatrix} \text{,} \; \mathbf x_3=\begin{pmatrix} -1\\ -1 \end{pmatrix} \text{,} \; \mathbf x_4=\begin{pmatrix} -1\\ +1 \end{pmatrix}\bigg \} The matrix \boldsymbol{i} has determinant +1 and gives a clockwise rotation of a quarter of a turn, since \mathbf x_2=\mathbf i \,\mathbf x_1\text{,}\;\mathbf x_3=\mathbf i \,\mathbf x_2\text{,}\;\mathbf x_4=\mathbf i \,\mathbf x_3\text{,}\; \mathbf x_1=\mathbf i \,\mathbf x_4 The matrix \boldsymbol{j} has determinant -1 and is the reflection about the first diagonal. It fixes the vertices \mathbf x_1 and \mathbf x_3 and reflects the vertices \mathbf x_2 and \mathbf x_4 into each other. The matrix \boldsymbol{k} is a reflection about the axis. It does \mathbf x_1 \leftrightarrow \mathbf x_2 and \mathbf x_3 \leftrightarrow \mathbf x_4 . The negative of the identity matrix is the square of the matrix \mathbf i , has determinant +1 , and rotates the square by 180° about the origin. It does \mathbf x_1 \leftrightarrow \mathbf x_3 and \mathbf x_2 \leftrightarrow \mathbf x_4 . N(q) = q q^* = w^2 + x^2 - y^2 - z^2, which is called the norm of the split-quaternion or the determinant of the associated matrix. 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 algebra. A split-quaternion with a nonzero norm has a multiplicative inverse, namely . In terms of matrices, this is equivalent to Cramer's 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 adjugate matrix by the determinant. The adjugate matrix is the transpose of the cofactor matrix and is also known as the adjoint or adjunct matrix. 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, \mathbb R), and the group of split quaternions of norm is isomorphic with \operatorname{SL}(2, \mathbb 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. j k = \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix}0 & i \\ -i & 0 \end{pmatrix} = \begin{pmatrix}-i & 0 \\ 0 & i \end{pmatrix} = - i . It follows 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 determinant form the special unitary group SU(1,1), which is used to describe hyperbolic motions of the Poincaré disk model in hyperbolic geometry. (a,b)(c,d)\ = \ (ac + d^* b, \ da + bc^* ) is used when producing the doubled product in the real-split cases. The doubled conjugate (a,b)^* = (a^*, - b), so that N(a,b) \ = \ (a,b)(a,b)^* \ = \ (a a^* - b b^* , 0). If a and b are split-complex numbers and split-quaternion q = (a,b) = ((w + z j), (y + xj)), then N(q) = a a^* - b b^* = w^2 - z^2 - (y^2 - x^2) = w^2 + x^2 - y^2 - z^2 . Let be a split-quaternion. Its real part is . Let be its nonreal part. One has , and therefore p^2=w^2+2wq-N(q). It follows that is a real number if and only is either a real number ( and ) or a purely nonreal split quaternion ( and ). The structure of the subalgebra \mathbb R[p] generated by follows straightforwardly. One has \mathbb R[p]=\mathbb R[q]=\{a+bq\mid a,b\in\mathbb R\}, and this is a commutative algebra. Its dimension is two except if is real (in this case, the subalgebra is simply \mathbb R ). The nonreal elements of \mathbb R[p] whose square is real have the form with a\in \mathbb R. Three cases have to be considered, which are detailed in the next subsections. With above notation, if q^2=0, (that is, if is nilpotent), then , that is, x^2-y^2-z^2=0. This implies that there exist and in \mathbb R such that and p=w+a\mathrm i + a\cos(t)\mathrm j + a\sin(t)\mathrm 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 \mathrm i + \cos(t)\mathrm j + \sin(t)\mathrm k form a circle; a subalgebra generated by a nilpotent element contains exactly one point of the circle; and the circle does not contain any other point. The algebra generated by a nilpotent element is isomorphic to \mathbb R[X]/\langle X^2\rangle and to the plane of dual numbers. This is the case where . Letting n=\sqrt{N(q)}, one has q^2 =-q^*q=N(q)=n^2=x^2-y^2-z^2. It follows that belongs to the hyperboloid of two sheets of equation x^2-y^2-z^2=1. Therefore, there are real numbers such that and p=w+n\cosh(u)\mathrm i + n\sinh(u)\cos(t)\mathrm j + n\sinh(u)\sin(t)\mathrm 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)\mathrm i + \sinh(u)\cos(t)\mathrm j + \sinh(u)\sin(t)\mathrm k form a hyperboloid of two sheets; a subalgebra generated by a split-quaternion with a nonreal part of positive norm contains exactly two opposite points on this hyperboloid, one on each sheet; and the hyperboloid does not contain any other point. The algebra generated by a split-quaternion with a nonreal part of positive norm is isomorphic to \mathbb R[X]/\langle X^2+1\rangle and to the field \Complex of complex numbers. This is the case where . Letting n=\sqrt{-N(q)}, one has q^2 = -q^*q=N(q)=-n^2=x^2-y^2-z^2. It follows that belongs to the hyperboloid of one sheet of equation . Therefore, there are real numbers such that and p=w+n\sinh(u)\mathrm i + n\cosh(u)\cos(t)\mathrm j + n\cosh(u)\sin(t)\mathrm 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)\mathrm i + \cosh(u)\cos(t)\mathrm j + \cosh(u)\sin(t)\mathrm k form a hyperboloid of one sheet; a subalgebra generated by a split-quaternion with a nonreal part of negative norm contains exactly two opposite points on this hyperboloid; and the hyperboloid does not contain any other point. The algebra generated by a split-quaternion with a nonreal part of negative norm is isomorphic to \mathbb R[X]/\langle X^2-1\rangle and to the ring of split-complex numbers. It is also isomorphic (as an algebra) to \mathbb R^2 by the mapping defined by (1,0)\mapsto \frac{1+X}2, \quad (0,1)\mapsto \frac{1-X}2. J K = \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix}0 & 1 \\ -1 & 0 \end{pmatrix} = \begin{pmatrix}-1 & 0 \\ 0 & 1 \end{pmatrix} = -I . 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. 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 Association. Hans Beck compared split-quaternion transformations to the circle-permuting property of Möbius transformations in 1910. The split-quaternion structure has also been mentioned briefly in the Annals of Mathematics.

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