Split-biquaternion explained
In mathematics, a split-biquaternion is a hypercomplex number of the form
where
w,
x,
y, and
z are
split-complex numbers and i, j, and k multiply as in the
quaternion group. Since each
coefficient w,
x,
y,
z spans two
real dimensions, the split-biquaternion is an element of an eight-dimensional
vector space. Considering that it carries a multiplication, this vector space is an
algebra over the real field, or an algebra over a ring where the split-complex numbers form the ring. This algebra was introduced by
William Kingdon Clifford in an 1873 article for the
London Mathematical Society. It has been repeatedly noted in mathematical literature since then, variously as a deviation in terminology, an illustration of the
tensor product of algebras, and as an illustration of the direct sum of algebras.The split-biquaternions have been identified in various ways by algebraists; see
below.
Modern definition
A split-biquaternion is ring isomorphic to the Clifford algebra Cl0,3(R). This is the geometric algebra generated by three orthogonal imaginary unit basis directions, under the combination rule
eiej=\begin{cases}
-1&i=j,\\
-ejei&i ≠ j
\end{cases}
giving an algebra spanned by the 8 basis elements, with (
e1e2)
2 = (
e2e3)
2 = (
e3e1)
2 = −1 and ω
2 = (
e1e2e3)
2 = +1.The sub-algebra spanned by the 4 elements is the
division ring of Hamilton's
quaternions, .One can therefore see that
where is the algebra spanned by, the algebra of the
split-complex numbers.Equivalently,
Split-biquaternion group
The split-biquaternions form an associative ring as is clear from considering multiplications in its basis . When ω is adjoined to the quaternion group one obtains a 16 element group
(×).
Module
Since elements of the quaternion group can be taken as a basis of the space of split-biquaternions, it may be compared to a vector space. But split-complex numbers form a ring, not a field, so vector space is not appropriate. Rather the space of split-biquaternions forms a free module. This standard term of ring theory expresses a similarity to a vector space, and this structure by Clifford in 1873 is an instance. Split-biquaternions form an algebra over a ring, but not a group ring.
Direct sum of two quaternion rings
The direct sum of the division ring of quaternions with itself is denoted
. The product of two elements
and
is
in this direct sum algebra.
Proposition: The algebra of split-biquaternions is isomorphic to
proof: Every split-biquaternion has an expression q = w + z ω where w and z are quaternions and ω2 = +1. Now if p = u + v ω is another split-biquaternion, their product is
The isomorphism mapping from split-biquaternions to
is given by
p\mapsto(u+v) ⊕ (u-v), q\mapsto(w+z) ⊕ (w-z).
In
, the product of these images, according to the algebra-product of
indicated above, is
This element is also the image of pq under the mapping into
Thus the products agree, the mapping is a homomorphism; and since it is
bijective, it is an isomorphism.
Though split-biquaternions form an eight-dimensional space like Hamilton's biquaternions, on the basis of the Proposition it is apparent that this algebra splits into the direct sum of two copies of the real quaternions.
Hamilton biquaternion
The split-biquaternions should not be confused with the (ordinary) biquaternions previously introduced by William Rowan Hamilton. Hamilton's biquaternions are elements of the algebra
Synonyms
The following terms and compounds refer to the split-biquaternion algebra:
- elliptic biquaternions –,
- Clifford biquaternion –,
- dyquaternions –
where
D =
split-complex numbers –,
, the direct sum of two quaternion algebras –
See also
References
- Book: Clifford, W. K.. W. K. Clifford. 1873. Mathematical Papers. Preliminary Sketch of Biquaternions. 195–197. Tucker. R..
- Book: Clifford, W. K.. W. K. Clifford. 1882. Mathematical Papers. The Classification of Geometric Algebras. 401. Tucker. R..
- P. R. . Girard . The quaternion group and modern physics . Eur. J. Phys. . 5 . 1 . 25–32 . 1984 . 10.1088/0143-0807/5/1/007 . 1984EJPh....5...25G . 250775753 .
- Book: Rooney, Joe . William Kingdon Clifford . Marco . Ceccarelli . Distinguished Figures in Mechanism and Machine Science: Their Contributions and Legacies . https://books.google.com/books?id=UmBnVMA5ri4C&pg=PA79 . 2007 . Springer . 978-1-4020-6366-4 . 79– .
- Book: Joly, Charles Jasper . Charles Jasper Joly . A Manual of Quaternions . 1905 . Macmillan . 21 .
- Book: Rosenfeld, Boris . Geometry of Lie Groups . Kluwer . 1997 . 978-0-7923-4390-5 . 48 .
- Book: Bourbaki, N. . Nicolas Bourbaki . Elements of the History of Mathematics . 1994 . J. . Meldrum. John D. P. Meldrum . 2013 . Springer . 978-3-642-61693-8 . 137 .
- Book: van der Waerden, B. L. . B. L. van der Waerden . A History of Algebra . registration . Springer . 1985 . 978-0-387-13610-3 . 188 .