In
abstract algebra, the split-quaternions or coquaternions form an
algebraic structure introduced by
James Cockle
Sir James Cockle FRS FRAS FCPS (14 January 1819 – 27 January 1895) was an English lawyer and mathematician.
Cockle was born on 14 January 1819. He was the second son of James Cockle, a surgeon, of Great Oakley, Essex. Educated at Charterh ...
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 has been 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.
Definition
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
:
They satisfy the same multiplication table as the corresponding split-quaternions. As these matrices form a basis of the two by two matrices, the
function that maps to
(respectively) induces an
algebra isomorphism
In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in ,
* F(kx) = kF( ...
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
dihedral group D
4, the
symmetry group of a square. In fact, if one considers a square whose vertices are the points whose coordinates are or , the matrix
is the clockwise rotation of the quarter of a turn,
is the symmetry around the first diagonal, and
is the symmetry around the axis.
Properties
Like the
quaternions introduced by
Hamilton Hamilton may refer to:
People
* Hamilton (name), a common British surname and occasional given name, usually of Scottish origin, including a list of persons with the surname
** The Duke of Hamilton, the premier peer of Scotland
** Lord Hamilto ...
in 1843, they form a four
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
real
associative algebra. But like the matrices and unlike the 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 of sign the two other entries.
The product of a split-quaternion with its conjugate is the
isotropic quadratic form:
:
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 means that split-quaternions and 2×2 matrices 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 matrix, this is
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 matrices shows that the multiplicative group of split-quaternions with a nonzero norm is isomorphic with
and the group of split quaternions of norm is isomorphic with
Representation as complex matrices
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
In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in ,
* F(kx) = kF( ...
that maps a split-quaternion to the matrix
:
Here, (
italic) is the
imaginary unit, which must not be confused with the basic split quaternion (
upright roman).
The image of this homomorphism is the
matrix ring formed by the matrices of the form
:
where the superscript
denotes a
complex conjugate.
This homomorphism maps respectively the split-quaternions on the matrices
:
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
:
Then, applied to the representation of split-quaternions as real matrices, the above algebra homomorphism is the matrix similarity.
:
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.
Generation from split-complex numbers
Split-quaternions may be generated by
modified Cayley-Dickson construction similar to the method of
L. E. Dickson and
Adrian Albert
Abraham Adrian Albert (November 9, 1905 – June 6, 1972) was an American mathematician. In 1939, he received the American Mathematical Society's Cole Prize in Algebra for his work on Riemann matrices. He is best known for his work on the A ...
. for the division algebras C, H, and O. The multiplication rule
is used when producing the doubled product in the real-split cases. The doubled conjugate
so that
If ''a'' and ''b'' are
split-complex numbers and split-quaternion
then
Stratification
In this section, the
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
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