Hypercomplex number

In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory.

In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature, added to the real and complex numbers. The concept of a hypercomplex number covered them all, and called for a discipline to explain and classify them.

It was matrix algebra that harnessed the hypercomplex systems. First, matrices contributed new hypercomplex numbers like 2 × 2 real matrices (see Split-quaternion). Soon the matrix paradigm began to explain the others as they became represented by matrices and their operations. In 1907 Joseph Wedderburn showed that associative hypercomplex systems could be represented by square matrices, or direct product of algebras of square matrices.[3][4] From that date the preferred term for a hypercomplex system became associative algebra as seen in the title of Wedderburn's thesis at University of Edinburgh. Note however, that non-associative systems like octonions and hyperbolic quaternions represent another type of hypercomplex number.

As Hawkins[5] explains, the hypercomplex numbers are stepping stones to learning about Lie groups and group representation theory. For instance, in 1929 Emmy Noether wrote on "hypercomplex quantities and representation theory".[6] In 1973 Kantor and Solodovnikov published a textbook on hypercomplex numbers which was translated in 1989.[7][8]

Karen Parshall has written a detailed exposition of the heyday of hypercomplex numbers,[9] including the role of mathematicians including Theodor Molien[10] and Eduard Study.[11] For the transition to modern algebra, Bartel van der Waerden devotes thirty pages to hypercomplex numbers in his History of Algebra.[12]

Theorem:[7]: 14, 15 [13][14] Up to isomorphism, there are exactly three 2-dimensional unital algebras over the reals: the ordinary complex numbers, the split-complex numbers, and the dual numbers. In particular, every 2-dimensional unital algebra over the reals is associative and commutative.

for some real numbers a0 and a1. Using the common method of completing the square by subtracting a1u and adding the quadratic complement a2
1
/ 4 to both sides yields

In a 2004 edition of Mathematics Magazine the 2-dimensional real algebras have been styled the "generalized complex numbers".[15] The idea of cross-ratio of four complex numbers can be extended to the 2-dimensional real algebras.[16]

Putting aside the bases which contain an element ei such that ei2 = 0 (i.e. directions in the original space over which the quadratic form was degenerate), the remaining Clifford algebras can be identified by the label Clp,q(R), indicating that the algebra is constructed from p simple basis elements with ei2 = +1, q with ei2 = −1, and where R indicates that this is to be a Clifford algebra over the reals—i.e. coefficients of elements of the algebra are to be real numbers.

These algebras, called geometric algebras, form a systematic set, which turn out to be very useful in physics problems which involve rotations, phases, or spins, notably in classical and quantum mechanics, electromagnetic theory and relativity.

Examples include: the complex numbers Cl0,1(R), split-complex numbers Cl1,0(R), quaternions Cl0,2(R), split-biquaternions Cl0,3(R), split-quaternions Cl1,1(R) ≈ Cl2,0(R) (the natural algebra of two-dimensional space); Cl3,0(R) (the natural algebra of three-dimensional space, and the algebra of the Pauli matrices); and the spacetime algebra Cl1,3(R).

The elements of the algebra Clp,q(R) form an even subalgebra Cl[0]
q+1,p
(R) of the algebra Clq+1,p(R), which can be used to parametrise rotations in the larger algebra. There is thus a close connection between complex numbers and rotations in two-dimensional space; between quaternions and rotations in three-dimensional space; between split-complex numbers and (hyperbolic) rotations (Lorentz transformations) in 1+1-dimensional space, and so on.

Whereas Cayley–Dickson and split-complex constructs with eight or more dimensions are not associative with respect to multiplication, Clifford algebras retain associativity at any number of dimensions.

In 1995 Ian R. Porteous wrote on "The recognition of subalgebras" in his book on Clifford algebras. His Proposition 11.4 summarizes the hypercomplex cases:[17]

For extension beyond the classical algebras, see Classification of Clifford algebras.

The first algebras in this sequence are the four-dimensional quaternions, eight-dimensional octonions, and 16-dimensional sedenions. An algebraic symmetry is lost with each increase in dimensionality: quaternion multiplication is not commutative, octonion multiplication is non-associative, and the norm of sedenions is not multiplicative.

The Cayley–Dickson construction can be modified by inserting an extra sign at some stages. It then generates the "split algebras" in the collection of composition algebras instead of the division algebras:

Unlike the complex numbers, the split-complex numbers are not algebraically closed, and further contain nontrivial zero divisors and non-trivial idempotents. As with the quaternions, split-quaternions are not commutative, but further contain nilpotents; they are isomorphic to the square matrices of dimension two. Split-octonions are non-associative and contain nilpotents.

The tensor product of any two algebras is another algebra, which can be used to produce many more examples of hypercomplex number systems.