Hypercomplex number
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.
The cataloguing project began in 1872 when Benjamin Peirce first published his Linear Associative Algebra, and was carried forward by his son Charles Sanders Peirce.
[1] Most significantly, they identified the nilpotent and the idempotent elements as useful hypercomplex numbers for classifications.
Hurwitz and Frobenius proved theorems that put limits on hypercomplexity: Hurwitz's theorem says finite-dimensional real composition algebras are the reals
, and the Frobenius theorem says the only real associative division algebras are
In 1958 J. Frank Adams published a further generalization in terms of Hopf invariants on H-spaces which still limits the dimension to 1, 2, 4, or 8.
For instance, 2 x 2 real matrices were found isomorphic to coquaternions.
Soon the matrix paradigm began to explain several others as they were represented by matrices and their operations.
[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.
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.
A technical approach to hypercomplex numbers directs attention first to those of dimension two.
Using the common method of completing the square by subtracting a1u and adding the quadratic complement a21 / 4 to both sides yields Thus
The three cases depend on this real value: The complex numbers are the only 2-dimensional hypercomplex algebra that is a field.
Split algebras such as the split-complex numbers that include non-real roots of 1 also contain idempotents
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.
Over the real numbers this is equivalent to being able to define a symmetric scalar product, u ⋅ v = 1/2(uv + vu) that can be used to orthogonalise the quadratic form, to give a basis {e1, ..., ek} such that:
Imposing closure under multiplication generates a multivector space spanned by a basis of 2k elements, {1, e1, e2, e3, ..., e1e2, ..., e1e2e3, ...}.
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(
), indicating that the algebra is constructed from p simple basis elements with ei2 = +1, q with ei2 = −1, and where
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.
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] All of the Clifford algebras Clp,q(
After the sedenions are the 32-dimensional trigintaduonions (or 32-nions), the 64-dimensional sexagintaquatronions (or 64-nions), the 128-dimensional centumduodetrigintanions (or 128-nions), the 256-dimensional ducentiquinquagintasexions (or 256-nions), and ad infinitum, as summarized in the table below.
As with the quaternions, split-quaternions are not commutative, but further contain nilpotents; they are isomorphic to the square matrices of dimension two.
