In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, 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 Charte ...
in 1849 under the latter name. They form an
associative algebra of dimension four over the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s.
After introduction in the 20th century of coordinate-free definitions of
rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
and
algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
, it has been proved that the algebra of split-quaternions is
isomorphic to the
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
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
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
, these relations imply
:,
:,
and also .
So, the split-quaternions form a
real vector space
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
of dimension four with as a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
. They form also a
noncommutative ring
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...
, by extending the above product rules by
distributivity
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmeti ...
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
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
that maps to
(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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
under this multiplication, which is
isomorphic to the
dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
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 Hamilt ...
in 1843, they form a four
dimensional real
associative algebra. But like the matrices and unlike the quaternions, the split-quaternions contain nontrivial
zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zer ...
s,
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
elements, and
idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
s. (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
In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadratic form on a vector s ...
:
:
which is called the
''norm'' of the split-quaternion or the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the associated matrix.
The real part of a split-quaternion is . It equals the
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''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
In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies
:N(xy) = N(x)N(y)
for all and in .
A composition algebra includes an involution ...
. 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
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/ ...
, namely . In terms of matrix, this is
Cramer rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of ...
that asserts that a matrix is
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
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
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
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
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, 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
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
.
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
In linear algebra, two ''n''-by-''n'' matrices and are called similar if there exists an invertible ''n''-by-''n'' matrix such that
B = P^ A P .
Similar matrices represent the same linear map under two (possibly) different bases, with being ...
. 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)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
. This is used for in
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
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
Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also reme ...
and
Adrian Albert. 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
subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
"Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operat ...
s 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