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 2×2 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 Cl2,0(R) ≅ Cl1,1(R) 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.

Definition

The split-quaternions are the linear combinations (with real coefficients) of four basis elements 1, i, j, k that satisfy the following product rules:

i2 = −1,

j2 = 1,

k2 = 1,

ij = k = −ji.

By associativity, these relations imply

jk = −i = −kj,

ki = j = −ik,

and also ijk = 1.

So, the split-quaternions form a real vector space of dimension four with {1, i, j, k} as a basis. They form also a noncommutative ring, by extending the above product rules by distributivity to all split-quaternions.

The square matrices

1

=

(

1

0

0

1

)

,

i

=

(

0

1

−

1

0

)

,

j

=

(

0

1

1

0

)

,

k

=

(

1

0

0

−

1

)

.