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Concept in mathematics
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
Hurwitz_quaternion_order
Generalization of Gaussian integers to quaternions
In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd
Hurwitz_quaternion
Generalization of quaternions to other fields
Composition algebra Cyclic algebra Octonion algebra Hurwitz quaternion order Hurwitz quaternion See Milies & Sehgal, An introduction to group rings,
Quaternion_algebra
German mathematician (1859–1919)
determinant Hurwitz-stable matrix Routh–Hurwitz matrix Hurwitz numbers Hurwitz polynomial Hurwitz problem Hurwitz quaternion order Hurwitz quaternion Hurwitz scheme
Adolf_Hurwitz
Four-dimensional number system
numbers, quaternion multiplication is not commutative, meaning that the result of multiplying two quaternions depends on their order. Quaternions can be
Quaternion
1009, 1025, 1459, 1537, 2091, ... (sequence A179982 in the OEIS) Hurwitz quaternion order Elkies, N.: Shimura curve computations. Algorithmic number theory
Hurwitz_surface
idea of order is still important, but the phenomena are different. For example, the Hurwitz quaternions form a maximal order in the quaternions with rational
Order_(ring_theory)
Non-abelian group of order eight
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset {
Quaternion_group
Compact Riemann surface of genus 3
} One chooses a suitable Hurwitz quaternion order Q H u r {\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }} in the quaternion algebra, Γ(I) is then the
Klein_quartic
Mathematical group
chooses a suitable Hurwitz quaternion order Q H u r {\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }} in the quaternion algebra. Here the order Q H u r {\displaystyle
(2,3,7)_triangle_group
Every natural number can be represented as the sum of four integer squares
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 (arithmetic
Lagrange's four-square theorem
Lagrange's_four-square_theorem
In mathematics, quaternions are a non-commutative number system that extends the complex numbers. Quaternions and their applications to rotations were
History_of_quaternions
Non-associative algebras with positive-definite quadratic form
complex numbers, the quaternions, or the octonions, and that there are no other possibilities. Such algebras, sometimes called Hurwitz algebras, are examples
Hurwitz's theorem (composition algebras)
Hurwitz's_theorem_(composition_algebras)
Type of group in group theory
taking the unit groups of orders in quaternion algebras over number fields (for example the Hurwitz quaternion order). Similar constructions can be performed
Arithmetic_group
principal congruence subgroups. Here the choices of quaternion algebra and Hurwitz quaternion order are described at the triangle group page. Choosing
MacBeath_surface
Three Riemann surfaces with same symmetry
maximal order of D {\displaystyle D} (see Hurwitz quaternion order), described explicitly by Noam Elkies [1]. In order to construct the first Hurwitz triplet
First_Hurwitz_triplet
Quaternion of norm 1 (unit quaternion)
include the twenty-four Hurwitz quaternions that have the norm 1 and form vertices of a 24-cell polychoron. Hamilton defined a quaternion as the quotient of
Versor
Method for producing composition algebras
with involution of twice the dimension. Hurwitz's theorem states that the reals, complex numbers, quaternions, and octonions are the only finite-dimensional
Cayley–Dickson_construction
Nonabelian group in algebraic group theory
article on quaternions and spatial rotations.) Explicitly, the binary tetrahedral group is given as the group of units in the ring of Hurwitz integers.
Binary_tetrahedral_group
Mathematics
_{g})\geq {\frac {4}{3}}\log g,} resulting from an analysis of the Hurwitz quaternion order. A similar bound holds for more general arithmetic Fuchsian groups
Systoles_of_surfaces
Type of algebras, possibly non associative
advanced the study of the Hurwitz problem with a survey of efforts to that date, and by exhibiting the method of doubling the quaternions to obtain Cayley numbers
Composition_algebra
Type of mathematical group
can be obtained as the surface associated to a particular order, the Hurwitz quaternion order, and it is compact of volume π / 21 {\displaystyle \pi /21}
Arithmetic_Fuchsian_group
tetrahedral group, 2T, consisting of the 24 Hurwitz units, forms a normal subgroup of index 2. The quaternion group, Q8, consisting of the 8 Lipschitz units
Binary_octahedral_group
Nonabelian group of order 120
algebra of quaternions, the binary icosahedral group is concretely realized as a discrete subgroup of the versors, which are the quaternions of norm one
Binary_icosahedral_group
Regular object in four dimensional geometry
Hurwitz integral quaternions. The vertices of the 24-cell form the group of units (i.e. the group of invertible elements) in the Hurwitz quaternion ring
24-cell
52-dimensional exceptional simple Lie group
center of the other). They form a ring called the Hurwitz quaternion ring. The 24 Hurwitz quaternions of norm 1 form the vertices of a 24-cell centered
F4_(mathematics)
In mathematics, a Riemann surface
have a realization as the order- 2 {\displaystyle 2} quotient of the group of norm- 1 {\displaystyle 1} elements of a quaternion algebra, but the ( 3 , 3
Bolza_surface
Mathematical operation on vectors in 3D space
quaternions. The nonexistence of nontrivial vector-valued cross products of two vectors in other dimensions is related to the result from Hurwitz's theorem
Cross_product
Hypercomplex number system
Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative
Octonion
Type of group in abstract algebra
Theo Douvropoulos; Joel Brewster Lewis; Alejandro H. Morales (2022), "Hurwitz Numbers for Reflection Groups I: Generatingfunctionology", Enumerative
Symmetric_group
Natural number
form, the same configuration may be identified with the 24 unit Hurwitz quaternions, which form the binary tetrahedral group. The optimal sphere packing
24_(number)
Mathematical concept
the ring of Hurwitz quaternions, and its automorphism group has order 24, contains a normal subgroup of order 8 isomorphic to the quaternion group, and
Supersingular_elliptic_curve
Sporadic simple group
can be realized as a Galois group over the rational numbers, and as a Hurwitz group. The monster is unusual among simple groups in that there is no known
Monster_group
Sporadic simple group
and Alexander Bromberg presented a complementary positive-dimensional Hurwitz-space approach to the inverse Galois problem for M23 over Q, beginning
Mathieu_group_M23
Sporadic simple group
multiplication tables, with w the same as a and w2 the same as 1 + a.) J2 is thus a Hurwitz group, a finite homomorphic image of the (2,3,7) triangle group. The matrix
Janko_group_J2
Sporadic simple group
a^{2}=b^{3}=(ab)^{7}=(abab^{-1})^{10}=1} J 1 {\displaystyle J_{1}} is thus a Hurwitz group, a finite homomorphic image of the (2,3,7) triangle group. Janko
Janko_group_J1
Ring without nonzero zero divisors
noncommutative subring of the quaternions, hence a noncommutative domain. Similarly, the set of all Hurwitz quaternions, that is, quaternions of the form a + b i
Domain_(ring_theory)
Algebraic variety
than subgroups of the modular group; a class of them constructed from quaternion algebras is also of interest in number theory. The covering X(N) → X(1)
Modular_curve
Relationship between certain vector spaces
Spin(8). Triple product, may be related to the 4-dimensional triality (on quaternions) John Frank Adams (1981), Spin(8), Triality, F4 and all that, in "Superspace
Triality
Length in a vector space
qq^{*}~}}={\sqrt {\,q^{*}q~}}={\sqrt {\,a^{2}+b^{2}+c^{2}+d^{2}~}}} for every quaternion q = a + b i + c j + d k {\displaystyle q=a+b\,\mathbf {i} +c\,\mathbf
Norm_(mathematics)
Arithmetical operation
multiplication is not, in general, commutative for matrices and quaternions. Hurwitz's theorem shows that for the hypercomplex numbers of dimension 8 or
Multiplication
Groups of point isometries in 3 dimensions
3D, are the quaternion group (of order 8), Z3 × Z3 (of order 9), the dicyclic group Dic3 (of order 12), and 10 of the 14 groups of order 16. The column
Point groups in three dimensions
Point_groups_in_three_dimensions
Orientation-preserving mapping class group of the torus
(2, 3, 7) triangle group (and associated tiling) is the cover for all Hurwitz surfaces. The group SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z}
Modular_group
group Aut(F4). The vertices of the 288-cell are precisely the 24 Hurwitz unit quaternions with norm squared 1, united with the 24 vertices of the dual 24-cell
Truncated_24-cells
Regular tessellation in 4D Euclidean space
Hurwitz quaternions with even square norm. The vertices of the honeycomb lie at the deep holes of the D4 lattice. These are the Hurwitz quaternions with
24-cell_honeycomb
Mathematical structure in abstract algebra
None of the three is a complex algebra. Hurwitz quaternions form a non-commutative *-ring with the quaternion conjugation. The matrix algebra of n × n
*-algebra
Algebraic curve in mathematics
Modularity theorem Moduli stack of elliptic curves Nagell–Lutz theorem Riemann–Hurwitz formula Wiles's proof of Fermat's Last Theorem Sarli, J. (2012). "Conics
Elliptic_curve
Left-invariant (or right-invariant) measure on locally compact topological group
1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in
Haar_measure
Complex number whose real and imaginary parts are both integers
Algebraic integer Cyclotomic field Eisenstein integer Eisenstein prime Hurwitz quaternion Proofs of Fermat's theorem on sums of two squares Proofs of quadratic
Gaussian_integer
Algorithm for computing greatest common divisors
mathematical objects, such as polynomials, quadratic integers and Hurwitz quaternions. In the latter cases, the Euclidean algorithm is used to demonstrate
Euclidean_algorithm
2003 mathematics text
be represented as sums of four squares (proved using the norms of Hurwitz quaternions), and quadratic reciprocity. Chapter 3 concerns group theory, and
Elementary Number Theory, Group Theory and Ramanujan Graphs
Elementary_Number_Theory,_Group_Theory_and_Ramanujan_Graphs
Number with a real and an imaginary part
reals, complex numbers, quaternions and octonions are all normed division algebras over R {\displaystyle \mathbb {R} } . By Hurwitz's theorem they are the
Complex_number
Russian mathematician (1861–1941)
of algebraic equations, number theory, projective geometry, theory of quaternions, history of mathematics and others. Some of these courses were new for
Theodor_Molien
algebras over the reals — the real numbers, the complex numbers and the quaternions. The only non-associative division algebra is the algebra of octonions
Exceptional_object
Geometric arrangements of points, foundational to Lie theory
half-integers is not allowed). This lattice is isomorphic to the lattice of Hurwitz quaternions. The root system G2 has 12 roots, which form the vertices of a hexagram
Root_system
all positive entries. Quaternionic matrix A matrix whose entries are quaternions. Random matrix A matrix whose entries are random variables Sign matrix
List_of_named_matrices
Open convex self-dual cones
Hn(H) be the space of self-adjoint n by n matrices with entries in the quaternions, inner product (a,b) = Re Tr ab* and Jordan product a ∘ b = 1/2(ab
Symmetric_cone
Mathematical ring whose elements are matrices
3.6(a) Lecture VII of Sir William Rowan Hamilton (1853) Lectures on Quaternions, Hodges and Smith Droste & Kuich (2009), p. 7 Droste & Kuich (2009),
Matrix_ring
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HURWITZ QUATERNION-ORDER
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HURWITZ QUATERNION-ORDER
HURWITZ QUATERNION-ORDER
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HURWITZ QUATERNION-ORDER
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