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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 ]
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.
One chooses a suitable Hurwitz quaternion order in the quaternion algebra, Γ(I) is then the group of norm 1 elements in +. The least absolute value of a trace of a hyperbolic element in Γ( I ) is η 2 + 3 η + 2 {\displaystyle \eta ^{2}+3\eta +2} , corresponding the value 3.936 for the systole of the Klein quartic, one of the highest in this ...
The ring of Hurwitz quaternions, also known as integral quaternions. A quaternion a = a 0 + a 1 i + a 2 j + a 3 k is integral if either all the coefficients a i are integers or all of them are half-integers. All free associative algebras. [1]
In mathematics, a versor is a quaternion of norm one (a unit quaternion).Each versor has the form = = + , =, [,], where the r 2 = −1 condition means that r is a unit-length vector quaternion (or that the first component of r is zero, and the last three components of r are a unit vector in 3 dimensions).