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The Hurwitz quaternions form an order (in the sense of ring theory) in the division ring of quaternions with rational components. It is in fact a maximal order ; this accounts for its importance. The Lipschitz quaternions, which are the more obvious candidate for the idea of an integral quaternion , also form an order.
Adolf Hurwitz (German: [ˈaːdɔlf ˈhʊʁvɪts]; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory. Early life [ edit ]
Hurwitz and Frobenius proved theorems that put limits on hypercomplexity: Hurwitz's theorem says finite-dimensional real composition algebras are the reals , the complexes , the quaternions , and the octonions , and the Frobenius theorem says the only real associative division algebras are , , and .
These may be interpreted as multiplicativity for the norms on the complex numbers (), quaternions (), and octonions (), respectively. [1]: 1–3 [2] The Hurwitz problem for the field K is to find general relations of the form
The Hurwitz quaternion order is a specific order in a quaternion algebra over a suitable number field. The order is of particular importance in Riemann surface theory, in connection with surfaces with maximal symmetry , namely the Hurwitz surfaces . [ 1 ]
Fascination with quaternions began before the language of set theory and mathematical structures was available. In fact, there was little mathematical notation before the Formulario mathematico. The quaternions stimulated these advances: For example, the idea of a vector space borrowed Hamilton's term but changed its meaning. Under the modern ...
Thus, the quaternion components ,,, are either all integers or all half-integers, depending on whether is even or odd, respectively. The set of Hurwitz quaternions forms a ring; that is to say, the sum or product of any two Hurwitz quaternions is likewise a Hurwitz quaternion.
The quaternion a + bi + cj + dk can be represented as the 2×2 complex matrix [ a + b i c + d i − c + d i a − b i ] . {\displaystyle {\begin{bmatrix}~~a+bi&c+di\\-c+di&a-bi\end{bmatrix}}.} This defines a map Ψ mn from the m by n quaternionic matrices to the 2 m by 2 n complex matrices by replacing each entry in the quaternionic matrix by ...