Split-quaternion

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Split-quaternion multiplication
× 1 i j k
1 1 i j k
i i −1 k −j
j j −k 1 −i
k k j i 1

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 has been 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.

Definition[edit]

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.

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 that maps 1, i, j, k 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 1, i, j, k, −1, −i, −j, −k form a group under this multiplication, which is isomorphic to the dihedral group D4, the symmetry group of a square. In fact, if one considers a square whose vertices are the points whose coordinates are 0 or 1, 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 x axis.

Properties[edit]

Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real associative algebra. But like the matrices and unlike the quaternions, the split-quaternions contain nontrivial zero divisors, nilpotent elements, and idempotents. (For example, 1/2(1 + j) is an idempotent zero-divisor, and i − j 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 isomorphim.

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 q = w + xi + yj + zk, is q = wxi − yj − zk. 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:

which is called the norm of the split-quaternion or the determinant of the associated matrix.

The real part of a split-quaternion q = w + xi + yj + zk is w = (q + q)/2. It equals the trace of associated matrix.

The norm of a product 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. 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, namely q/N(q). In terms of matrix, this is Cramer rule that asserts that a matrix is invertible 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 1 is isomorphic with

Representation as complex matrices[edit]

There is a representation of the split-quaternions as a unital associative subalgebra of the 2×2 matrices with complex entries. This representation can be defined by the algebra homomorphism that maps a split-quaternion w + xi + yj + zk to the matrix

Here, i (italic) is the imaginary unit, which must not be confused with the basic split quaternion i (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.

This homomorphism maps respectively the split-quaternions i, j, k 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 2×2 real matrices, and using matrix similarity. Let S be the matrix

Then, applied to the representation of split-quaternions as 2×2 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 1 are exactly the elements of the special unitary group SU(1,1). This is used for in hyperbolic geometry for describing hyperbolic motions of the Poincaré disk model.[1]

Generation from split-complex numbers[edit]

Kevin McCrimmon [2] has shown how all composition algebras can be constructed after the manner promulgated by L. E. Dickson and Adrian Albert for the division algebras C, H, and O. Indeed, he presents the multiplication rule

to be used when producing the doubled product in the real-split cases. As before, the doubled conjugate so that

If a and b are split-complex numbers and split-quaternion

then

Stratification[edit]

In this section, the subalgebras generated by a single split-quaternion are studied and classified.

Let p = w + xi + yj + zk be a split-quaternion. Its real part is w = 1/2(p + p*). Let q = pw = 1/2(pp*) be its nonreal part. One has q* = –q, and therefore It follows that is a real number if and only p is either a real number (q = 0 and p = w) or a purely nonreal split quaternion (w = 0 and p = q).

The structure of the subalgebra generated by p follows straightforwardly. One has

and this is a commutative algebra. Its dimension is two except if p is real (in this case, the subalgebra is simply ).

The nonreal elements of whose square is real have the form aq with

Three cases have to be considered, which are detailed in the next subsections.

Nilpotent case[edit]

With above notation, if (that is, if q is nilpotent), then N(q) = 0, that is, This implies that there exist w and t in such that 0 ≤ t < 2π and

This is a parametrization of all split-quaternions whose nonreal part is nilpotent.

This is also a parameterization of these subalgebras by the points of a circle: the split-quaternions of the form form a circle; a subalgebra generated by a nilpotent element contains exactly one point of the circle; and the circle does not contain any other point.

The algebra generated by a nilpotent element is isomorphic to and to the space of the dual numbers.

Decomposable case[edit]

This is the case where N(q) > 0. Letting one has

It follows that 1/n q belongs to the hyperboloid of two sheets of equation Therefore, there are real numbers n, t, u such that 0 ≤ t < 2π and

This is a parametrization of all split-quaternions whose nonreal part has a positive norm.

This is also a parameterization of the corresponding subalgebras by the pairs of opposite points of a hyperboloid of two sheets: the split-quaternions of the form form a hyperboloid of two sheets; a subalgebra generated by a split-quaternion with a nonreal part of positive norm contains exactly two opposite points on this hyperboloid, one on each sheet; and the hyperboloid does not contain any other point.

The algebra generated by a split-quaternion with a nonreal part of positive norm is isomorphic to and to the space of the split-complex numbers. It is also isomorphic (as an algebra) to by the mapping defined by

Indecomposable case[edit]

Hyperboloid of one sheet
(the vertical axix is called x in the article)

This is the case where N(q) < 0. Letting one has

It follows that 1/n q belongs to the hyperboloid of one sheet of equation Therefore, there are real numbers n, t, u such that 0 ≤ t < 2π and

This is a parametrization of all split-quaternions whose nonreal part has a negative norm.

This is also a parameterization of the corresponding subalgebras by the pairs of opposite points of a hyperboloid of one sheet: the split-quaternions of the form form a hyperboloid of one sheet; a subalgebra generated by a split-quaternion with a nonreal part of negative norm contains exactly two opposite points on this hyperboloid; and the hyperboloid does not contain any other point.

The algebra generated by a split-quaternion with a nonreal part of negative norm is isomorphic to and to field of the complex numbers.

Stratification by the norm[edit]

As seen above, the purely nonreal split-quaternions of norm –1, 1 and 0 form respectively a hyperboloid of one sheet, a hyporboloid of two sheets and a circular cone in the space of non real quaternions.

These surfaces are pairwise asymptote and do not intersect. Their complement consist of six connected regions:

  • the two regions located on the concave side of the hyperboloid of two sheets, where
  • the two regions between the hyperboloid of two sheets and the cone, where
  • the region between the cone and the hyperboloid of one sheet where
  • the region outside the hyperboloid of one sheet, where

This stratification can be refined by considering split-quaternions of a fixed norm: for every real number n ≠ 0 the purely nonreal split-quaternions of norm n form an hyperboloid. All these hyperboloids are asymptote to the above cone, and none of these surfaces intersect any other. As the set of the purely nonreal split-quaternions is the disjoint union of these surfaces, this provides the desired stratification.

Historical notes[edit]

The coquaternions were initially introduced (under that name)[3] in 1849 by James Cockle in the London–Edinburgh–Dublin Philosophical Magazine. The introductory papers by Cockle were recalled in the 1904 Bibliography[4] of the Quaternion Society. Alexander Macfarlane called the structure of split-quaternion vectors an exspherical system when he was speaking at the International Congress of Mathematicians in Paris in 1900.[5]

The unit sphere was considered in 1910 by Hans Beck.[6] For example, the dihedral group appears on page 419. The split-quaternion structure has also been mentioned briefly in the Annals of Mathematics.[7][8]

Synonyms[edit]

  • Para-quaternions (Ivanov and Zamkovoy 2005, Mohaupt 2006) Manifolds with para-quaternionic structures are studied in differential geometry and string theory. In the para-quaternionic literature k is replaced with −k.
  • Exspherical system (Macfarlane 1900)
  • Split-quaternions (Rosenfeld 1988)[9]
  • Antiquaternions (Rosenfeld 1988)
  • Pseudoquaternions (Yaglom 1968[10] Rosenfeld 1988)

See also[edit]

Notes[edit]

  1. ^ Karzel, Helmut & Günter Kist (1985) "Kinematic Algebras and their Geometries", in Rings and Geometry, R. Kaya, P. Plaumann, and K. Strambach editors, pp. 437–509, esp 449,50, D. Reidel ISBN 90-277-2112-2
  2. ^ Kevin McCrimmon (2004) A Taste of Jordan Algebras, page 64, Universitext, Springer ISBN 0-387-95447-3 MR2014924
  3. ^ James Cockle (1849), On Systems of Algebra involving more than one Imaginary, Philosophical Magazine (series 3) 35: 434,5, link from Biodiversity Heritage Library
  4. ^ A. Macfarlane (1904) Bibliography of Quaternions and Allied Systems of Mathematics, from Cornell University Historical Math Monographs, entries for James Cockle, pp. 17–18
  5. ^ Alexander Macfarlane (1900) Application of space analysis to curvilinear coordinates Archived 2014-08-10 at the Wayback Machine, Proceedings of the International Congress of Mathematicians, Paris, page 306, from International Mathematical Union
  6. ^ Hans Beck (1910) Ein Seitenstück zur Mobius'schen Geometrie der Kreisverwandschaften, Transactions of the American Mathematical Society 11
  7. ^ A. A. Albert (1942), "Quadratic Forms permitting Composition", Annals of Mathematics 43:161 to 77
  8. ^ Valentine Bargmann (1947), "Irreducible unitary representations of the Lorentz Group", Annals of Mathematics 48: 568–640
  9. ^ Rosenfeld, B.A. (1988) A History of Non-Euclidean Geometry, page 389, Springer-Verlag ISBN 0-387-96458-4
  10. ^ Isaak Yaglom (1968) Complex Numbers in Geometry, page 24, Academic Press

Further reading[edit]