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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.
The field of complex numbers is also isomorphic to three subsets of quaternions.) [22] A quaternion that equals its vector part is called a vector quaternion. The set of quaternions is a 4-dimensional vector space over the real numbers, with {,,,} as a basis, by the component-wise addition
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 ]
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 ]
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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.
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