Hamiltonian quaternions

Hamiltonian quaternions

Definition of

H

We define a unital

associative algebra

ℍ

over

ℝ

, of

dimension

4,
by the basis

{

𝟏

,

𝐢

,

𝐣

,

𝐤

}

and
the multiplication table

1

i

j

k

i

-

1

k

-

j

j

-

k

-

1

i

k

j

-

i

-

1

(where the

element

in row

x

and column

y

is

x

⁢

y

, not

y

⁢

x

).
Thus an arbitrary element of

ℍ

is of the form

a

⁢

𝟏

+

b

⁢

𝐢

+

c

⁢

𝐣

+

d

⁢

𝐤

,

a

,

b

,

c

,

d

∈

ℝ

(sometimes denoted by

⟨

a

,

b

,

c

,

d

⟩

or by

a

+

⟨

b

,

c

,

d

⟩

) and the

product

of two elements

⟨

a

,

b

,

c

,

d

⟩

and

⟨

α

,

β

,

γ

,

δ

⟩

(order matters)
is

⟨

w

,

x

,

y

,

z

⟩

where

w

=

a

⁢

α

-

b

⁢

β

-

c

⁢

γ

-

d

⁢

δ

x

=

a

⁢

β

+

b

⁢

α

+

c

⁢

δ

-

d

⁢

γ

y

=

a

⁢

γ

-

b

⁢

δ

+

c

⁢

α

+

d

⁢

β

z

=

a

⁢

δ

+

b

⁢

γ

-

c

⁢

β

+

d

⁢

α

The elements of

ℍ

are known as

Hamiltonian quaternions

.

Clearly the

subspaces

of

ℍ

generated by

{

𝟏

}

and by

{

𝟏

,

𝐢

}

are

subalgebras

isomorphic

to

ℝ

and

ℂ

respectively.

ℝ

is customarily identified with
the corresponding subalgebra of

ℍ

.
(We shall see in a moment that there are other and less obvious

embeddings

of

ℂ

in

ℍ

.)
The real numbers commute with all the elements of

ℍ

, and we have

λ

⋅

⟨

a

,

b

,

c

,

d

⟩

=

⟨

λ

⁢

a

,

λ

⁢

b

,

λ

⁢

c

,

λ

⁢

d

⟩

for

λ

∈

ℝ

and

⟨

a

,

b

,

c

,

d

⟩

∈

ℍ

.

Norm,

conjugate

, and

inverse

of a quaternion

Like the

complex numbers

(

ℂ

), the quaternions have a
natural

involution

called the

quaternion conjugate

. If

q

=

a

⁢

𝟏

+

b

⁢

𝐢

+

c

⁢

𝐣

+

d

⁢

𝐤

, then the quaternion
conjugate of

q

, denoted

q

¯

, is simply

q

¯

=

a

⁢

𝟏

-

b

⁢

𝐢

-

c

⁢

𝐣

-

d

⁢

𝐤

.

One can readily verify that if

q

=

a

⁢

𝟏

+

b

⁢

𝐢

+

c

⁢

𝐣

+

d

⁢

𝐤

,
then

q

⁢

q

¯

=

(

a

2

+

b

2

+

c

2

+

d

2

)

⁢

𝟏

.
(See

Euler four-square identity

.)
This product is used to form a norm

∥

⋅

∥

on the

algebra

(or the ring)

ℍ

: We define

∥

q

∥

=

s

where

q

⁢

q

¯

=

s

⁢

𝟏

.

If

v

,

w

∈

ℍ

and

λ

∈

ℝ

, then

1.

∥

v

∥

≥

0

with equality only if

v

=

⟨

0

,

0

,

0

,

0

⟩

=

0

2.

∥

λ

⁢

v

∥

=

|

λ

|

⁢

∥

v

∥

3.

∥

v

+

w

∥

≤

∥

v

∥

+

∥

w

∥

4.

∥

v

⋅

w

∥

=

∥

v

∥

⋅

∥

w

∥

which means that

ℍ

qualifies as a normed algebra when we give it the
norm

∥

⋅

∥

.

Because the norm of any nonzero quaternion

q

is real and nonzero,
we have

q

⁢

q

¯

∥

q

∥

2

=

q

¯

⁢

q

∥

q

∥

2

=

⟨

1

,

0

,

0

,

0

⟩

which shows that any nonzero quaternion has an inverse:

q

-

1

=

q

¯

∥

q

∥

2

.

Other embeddings of

C

into

H

One can use any non-zero

q

to define an embedding of

ℂ

into

ℍ

. If

𝐧

⁢

(

z

)

is a

natural embedding

of

z

∈

ℂ

into

ℍ

, then the embedding:

z

→

q

⁢

𝐧

⁢

(

z

)

⁢

q

-

1

is also an embedding into

ℍ

.
Because

ℍ

is an associative algebra, it is obvious
that:

(

q

⁢

𝐧

⁢

(

a

)

⁢

q

-

1

)

⁢

(

q

⁢

𝐧

⁢

(

b

)

⁢

q

-

1

)

=

q

⁢

(

𝐧

⁢

(

a

)

⁢

𝐧

⁢

(

b

)

)

⁢

q

-

1

and with the

distributive laws

, it is easy to check that

(

q

⁢

𝐧

⁢

(

a

)

⁢

q

-

1

)

+

(

q

⁢

𝐧

⁢

(

b

)

⁢

q

-

1

)

=

q

⁢

(

𝐧

⁢

(

a

)

+

𝐧

⁢

(

b

)

)

⁢

q

-

1

Rotations

in 3-space

Let us write

U

=

{

q

∈

ℍ

:

||

q

||

=

1

}

With

multiplication

,

U

is a group.
Let us briefly sketch the

relation

between

U

and the group

S

⁢

O

⁢

(

3

)

of rotations (about the origin) in 3-space.

An arbitrary element

q

of

U

can be expressed

cos

⁡

θ

2

+

sin

⁡

θ

2

⁢

(

a

⁢

𝐢

+

b

⁢

𝐣

+

c

⁢

𝐤

)

,
for some real numbers

θ

,

a

,

b

,

c

such that

a

2

+

b

2

+

c

2

=

1

.
The

permutation

v

↦

q

⁢

v

of

U

thus gives rise to a permutation
of the real sphere. It turns out that that permutation is a rotation.
Its axis is the line through

(

0

,

0

,

0

)

and

(

a

,

b

,

c

)

, and the angle
through which it rotates the sphere is

θ

.
If rotations

F

and

G

correspond to quaternions

q

and

r

respectively, then clearly the permutation

v

↦

q

⁢

r

⁢

v

corresponds
to the

composite

rotation

F

∘

G

.
Thus this

mapping

of

U

onto

S

⁢

O

⁢

(

3

)

is a

group homomorphism

.
Its kernel is the subset

{

1

,

-

1

}

of

U

, and thus it comprises
a double cover of

S

⁢

O

⁢

(

3

)

. The kernel has a geometric

interpretation

as well: two

unit vectors

in opposite

directions

determine the same
axis of rotation.

On the

algebraic

side, the quaternions provide an example of a

division ring

that is not a field.

Title

Hamiltonian quaternions

Canonical name

HamiltonianQuaternions

Date of creation

2013-03-22 12:35:42

Last modified on

2013-03-22 12:35:42

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Definition

Classification

msc 16W99

Synonym

quaternion

Related topic

EulerFourSquareIdentity

Related topic

QuaternionGroup

Related topic

HyperkahlerManifold

Related topic

MathematicalBiology

Defines

quaternion algebra

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