I
Quaternions
1.1 The
Quaternions
The Hamiltonian
symbols i, j,
quaternions
i2 ii
--
multiplication
The
non-z...
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I
Quaternions
1.1 The
Quaternions
The Hamiltonian
symbols i, j,
quaternions
i2 ii
--
multiplication
The
non-zero
H
the unitary
are
R-algebra generated by
-ii
=
k,
j2
=
ik
=
k2
_1'
=
-kj
=
is associative but
element has
4-dimensional division
ki
i)
=
obviously
multiplicative algebra over the
a
a
=
ao +
ali
not
-ik
commutative, and each a skew-field, and a
reals. Frobenius showed in 1877 that
a2i
+
=
inverse: We have
R, C and H are in fact the only finite-dimensional ciative and have no zero-divisors. For the element
we
the
k with the relations
a3k,
+
R-algebras
al C-
that
are asso-
(1.1)
R,
define a:= ao
ali
-
-
a2i
-
a3k,
Rea:= ao, Ima:= ali + Note
a2i
+
a3k.
that, in contrast with the complex numbers, conjugation obeys
Im
a
is not
a
real
number,
and that
Wb_
=
b a.
identify the real vector space H in the obvious subspace of purely imaginary quaternions with R3:
We shall the
W
=
way with
R ,
and
IMH.
embedding of the complex numbers i,-j,k equally qualify for the complex in and fact any.purely imaginary quaternion of square -1 imaginary unit, would do the job. Rom now on, however, we shall usually use the subfield C C ffff generated by 1, i.
The reals
are
identified with RI. The
C is less canonical. The quaternions
F. E. Burstall et al.: LNM 1772, pp. 1 - 4, 2002 © Springer-Verlag Berlin Heidelberg 2002
2
1
Quaternions
Occasionally be written
we
shall need the
Euclidean
inner
product
on
R4 which
can
as
< a, b >R=
Re(ab)
Re(ab)
=
2
(ab + ba).
We define
a
a,
>R
a
=
vfa- d.
Then
jabj A closer
study of
the quaternionic
jal Ibl.
=
(1.2)
multiplication displays
nice
geometric
as-
pects. We first mention that the quaternion multiplication incorporates both the products on V. In fact, using the representation (1.1)
usual vector and scalar
finds for a, b E Im Eff
one
=
R'
ab=axbAs
a
consequence
we
we
have
ba
if and only if Im a and Im b are linearly dependent over the reals. particular, the reals are the only quaternions that commute with all
=
In
(1-3)
state
Lemma 1. For a, b G H 1. ab
R-
others. 2. a' a
=
-1
if and only if Jal two-sphere
=
1 and
a
=
Im
a.
Note that the set
of
all such
is the usual
Proof.
Write
a
=
ao +
a', b
=
S2
C
bo
+
Y,
V
=
IMH.
where the prime denotes the
imaginary
part. Then ab
a'b'
=
aobo
+
aob'
+
a'bo
+
=
aobo
+
aob'
+
a'bo
+ a'
x
Y-