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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 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 ]
Let A be the set of quaternions of the form a + b i + c j + d k where a, b, c, and d are either all integers or all half-integers. The set A is a ring (in fact a domain) and a lattice and is called the ring of Hurwitz quaternions.
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The (2,3,7) triangle group admits a presentation in terms of the group of quaternions of norm 1 in a suitable order in a quaternion algebra. More specifically, the triangle group is the quotient of the group of quaternions by its center ±1. Let η = 2cos(2π/7). Then from the identity
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 .
perhaps making notes as you read. When you've finished, set aside three hours and write your answers to the questions in Part Three. Whatever your choice, enjoy the journey! THE TURNING POINT The idea started on New Year’s Day in 1980, when my boyfriend (now my husband), Tim, and I woke up in our flat in London. We’d been working
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